And does the Biltmore Estate qualify as either one?
A castle to me always implies a fortified building. So your link does not bring castles to mind in my opinion.
A castle is a fortification.
A palace is the home of a head of state.
The Biltmore mansion, while grander than many castles and palaces, is neither.
It may be petty semantics but I always thought that castles were to protect and defend people and things while palaces are merely giant and grand estates. A moat might differentiate the two although that is probably neither necessary nor sufficient.
However, you are probably right that there isn’t a clear distinction. Newport, RI has a whole row of places like the Biltmore built by very rich people trying to outdo one another. The Loire Valley in France (the Valley of Castles) probably has the most elaborate and biggest concentration of castles that you have ever laid eyes on but they were built for basically the same reason as those in Newport (rich ultra-escapes). Those are definitely what people think of as castles.
There are a few castle-like buildings in the U.S. The Boston area has some that eccentric people built and serve as celebration venues now.
It is probably a combination of function, history, architecture, and location and there is no fine line.
It could be defined as a palace. One of the definitions of a palace is a large and stately mansion or building. Palace in generally used to describe the residence of royalty such as a king or queen. The primary definition of castle is a walled or fortified residence of a prince or noble. Castle can also be used to describe a large house or mansion that has the elements of a medieval castle such as high walls or towers. But the name includes the usual American name for such a place, estate. The definition of estate is a piece of landed property that may include a large and elaborate house.
I strongly disagree in an educated way. Again, the Loire Valley of France is my main piece of evidence. The whole area is filled with glorious, self-described castles and many of them meet both of your criteria while some of them none. They are exactly what anyone would call castles (many actual examples of Disney style castles).
From a French Tourism website (more info and photos in it):
“This is the “country of thousand castles” : not only Royal Castles and Renaissance Palaces ,National Museum and Large magnificent Châteaux , but also : Manors ,Middle Age‘s Castles and Fortresses , small “Fairy tales” Manors ,mansions , Private Castles ,Châteaux Hotels ,“Self Catering” Castles , iscover castles, castles ,castles …more castles.”
To me, it just ain’t a castle if it ain’t castellated.
Rocks vs Marble 
My take on it is that the issue is usage drift. Originally, it seems clear: Middle Ages, a castle was a defensive fortification, and a palace was, well, a palatial residence. But as time went by and on the one hand the nobles dwelling at the castles began fancying them up to show off wealth and on the other hand weapons technology made them less effective as practical strongholds, many castle-owners began making them more and more palatial and less and less militarily practical.
In the 1600s Vauban developed “modern” fortification, and the classic tall-walled castle became obsolete (for most uses) as a primary military installation. By then kings relied standing armies rather than on the local Lords for real warfighting, so the still-militarily-practical strongholds became forts, manned by army personnel, and the remaining castes that were a nobleman’s residence were demilitarized. For the nobles who stayed in residence at old-style castles, these became fancy but thick-walled country homes, which often were remodelled to the point of losing military capability. But by tradition, these were still called “castles” – it’s what they had been for a couple of centuries prior.
After that time, you had the phenomenon of members of the nobility (or bourgeois arrivistes) building, either as replacement for the old castle or as brand new housing, structures that were really country-palaces or country-manors with little tactical usefulness (other than being solidly built and on high ground), but which they named “castle” instead of “palace” because (a) they were in countryside or in small towns rather than in the big city, (b) they did not want to upstage the royal palace by using the same word, © they incorporate architectural elements such as crenelations or turrets or cloisters, (d) they wanted to evoke nobility and chivalry and of course (e) the solidly built structure on high ground in the county where Mr. Big lived or the king spent the summer had been “the castle” for however many centuries and they liked it that way.
I would call the Biltmore Estate a palace. It’s more than a mansion or a manor, and the word “estate” doesn’t begin to describe the grandeur of the Biltmore. A few years back, the A&E cable network had a series called “America’s Castles,” and the Biltmore was included.
Interesting that my brother-in-law (wife’s brother), his wife, and I were discussing exactly this point on Monday evening – not specifically with reference to the Biltmore Estate (though my wife and her brother are from Asheville) but to the “castles” in the Loire Valley, which they propose to visit this summer. I held out for a definition of “castle” in which defensive fortification was the distinguishing feature. We settled on “palace” as the appropriate term for big honkin’ residences that may or may look like castles and may or may not have been home to nobility, royalty, or merely the stinking rich.
Which were called, with entirely unconvincing offhanded casualness by their owners, “cottages.”
IMO, JRDelirious has the right take on it. Castles were, from the Dark Ages through the time of Vauban, constructions that served as strong points for controlling a territory and as defensive structures for the inhabitants in case of invasion.
In the time of Louis XIV, and later in the Victorian era, people built them out of a desire to impress and a sense of nostalgia for the romantic past. So they’re castles by visual impression and by definition, even though they didn’t have the same function.
There were some non-heads of state who built palaces, by the way, though in general they were purposefully the large formal dwelling in which the head of state performed his/her ceremonial duties as head of state. Elizabeth II’s present primary residence, Buckingham Palace, was originally built by the Dukes of Buckingham. Blenheim Palace was the seat of the Dukes of Marlborough. The key point, however, is that palaces are in general not fortified, while castles generally are.
As a note, a castle sensu stricto is not a single interior dwelling, but a castellated wall enclosing an interior area, with at least one fortified tower, the keep, included within it. But typically a castle was a wall enclosing interior open ground and structures.
The Hearst “Castle” in San Simeon CA is a very beautiful place but cannot, I think, be called a castle. A palace? Maybe. An estate? I don’t know.
Anyone been there? Opinions?
Just went there two weeks ago, actually.
There’s nothing castle-like about Hearst Castle save the name. It’s definitely palatial, though! 
A very beautiful place - but to me, castle implies fortification or at least the imitation of fortification, and Hearst Castle has much more in common with European churches (of all things) than European castles.
[Former Loire Valley tour guide[sup][/sup] here.]*
I’d call the Biltmore Estate a Château (although I might drop the circonflex accent). Although the word’s obviously from the French, I’d say it has a usage in English that fits the middle-ground where neither castle nor palace have just the right feeling. Since the Biltmore Estate was consciously patterned after several of the Loire Valley châteaux (most notably the Château de Blois, it seems particularly appropriate in this case. [However, Château de Blois – unlike Biltmore – was built as a fortified location, so the original is also a “castle”.]
IMHO, a castle has to involve some elements of fortification, even if only in architectural touches. If it was built to repel invaders, it’s a castle. If it has crenellations – or a moat with a drawbridge – it’s probably a castle even if it’s only ever repelled Girl Scout groups who show up for a tour five minutes before closing time. If it’s pure “wedding-cake” architecture, like Chambord, IMHO it’s not a castle.
A palace denotes an official or semi-official residence of some person(s) with a high noble rank or public title. Thus, in the UK one finds Buckingham Palace (residence of the Monarch), Lambeth Palace (official London residence of the Archbishop of Canterbury), and the Palace of Westminster (aka Houses of Parliament, the official “home” of the group of people who make up Parliament, even if none of the individual members actually live there). By that definition, Biltmore couldn’t be a “palace” because it’s always been a private house – never associated with any official position, and of course the US doesn’t have ranks of nobility. On the other hand, the White House could be described as a “palace”, (as could various Governors’ Mansions and Mayoral residences), although given the circumstances under which the US became an independent nation, it’s rather unlikely to happen any time soon…
[[sup]*[/sup][Personal hijack]: When living in the Loire Valley, I undertook a fair amount of translation work and several French-English interpreter gigs. One of my clients was the local Tourist Board, and I had a major fight with them about the global use of the term “castle” in the English-language brochure that I was hired to help them re-write. They had settled on it because “that’s what’s in the dictionary” for château. I presented them with numerous cites of English-language publications using “château”, together with photos of castles, palaces, and – yes – châteaux. They were still skeptical, but when I pointed out that the English version of the relevant Michelin Green Guide was called "Châteaux of the Loire", it was a slam dunk. In France, you don’t argue with the Guide vert Michelin. (I suspect that the people who created the site linked to by Shagnasty just used a dictionary). Of course, several of the Loire Valley châteaux are also castles.
The Tourist Board later hired me as an interpreter for English-language organized VIP tours, and one day when the French tour guide had a sore throat I just ran the tours myself under her silent supervision, after which they let do them without her. Fun times…[/hijack]]
Antonius, an outstanding answer. Thank you!
Of course, this obliges me to mention the punster friend whose pet cats drank out of a water dish labelled “Chateau” 
A fine answer indeed. (The girl scout link was a bit much, but still…)
Excellent answer but worth mentioning that the Palace of Westminster is so called because it was originally where the King lived - a safe distance from the London mob!. When Henry VIII moved out in 1512 (to the Palace of Whitehall) it officially remained a Royal Palace although it developed as the home of Parliament.
As your link indicates, in its early days the White House was sometimes referred to as the “President’s Palace,” and its design was based on a ducal palace in Dublin.
In Latin America and many other places the president’s house is usually called the Presidential Palace. In Panama the presidential residence is the Palacio de las Garzas, “Palace of the Herons,” from the herons that live at a fountain in a courtyard.