What’s the difference? Are they interchangable?
Also, should you ever start a sentence with “Unfortunately”?
Continually refers to something that occurs repeatedly over a long period of time. It’s something that you habitually but not necessarily all at once.
Continuously refers to something that is uninterrupted or unbroken.
For example, one could sleep continuously for 12 hours. One could also continually nap every afternoon for an hour.
I’m not sure about the “Unfortunately” thing. I use it all of the time, and I’ve never heard anything referring to it being misused as such.
They are not interchangable at all.
Continually means that it happens all of the time.
Example: I am continually harassed by telemarketers while I am eating dinner.
Continuously means that it an action or an object is uninterrupted.
Example 1: That line of people stetches continuously all the way around the block.
Example 2: A baby screamed continuously for over 20 minutes on a flight I was on last week.
You can start a sentence with “Unfortunately” if you need to. It generally needs a comma after it but there is nothing grammatically incorrect about using it in that way. It can become a style issue if you do it continually. 
Continually doesn’t really mean frequently, but rather iteratively in all applicable cases. Continuously means at all times, in a single process. Unfortunately, (;)) both are generally used as if they meant often.
Tris
‘Continually’ = happens regularly or on an ongoing basis
‘Continuously’ = without interruption
There is no reason not to start a sentence with ‘Unfortunately’. It is an adverb, and would normally start an adverbial clause of mood.
Unfortunately, there are pedants who object to what are called sentence adverbs, that is adverbs that modify the whole sentence. They are doubtless the same ones who think that prepositions are bad words to end sentences with. They also don’t want you to carelessly split an infinitive. They are all ignoramuses who are trying to impose grammatical rules from some other language (usually Latin) onto English. Ignore them.
Wow, grammar mavens in flocks, and they all agree!
Surely the end is near.
Tris
That’s not a grammar question; it’s a style question.
Fortunately, there are a few adverbs that can be used to start a sentence and which give the opinon of the speaker. Thankfully, there is no rule against using “unfortunately.” (See example #2 from English Grammar Online – it’s a "what’s wrong with the sentence exercise, and none of the choices mention that using “unfortunately” is wrong).
Obviously, this is a backlash against using “Hopefully” in the manner. Unfortunately, the grammar reactionaries may argue this perfectly useful construction is “wrong.” Hopefully, common sense will prevail and people will use the words without fear of pedants.
I dunno about “the end,” but perhaps the end of GQ mods – next thing you know, the gun control (pro an con) people will answer General Questions with a General Answer. And then we’ll all be out on the streets in MPSIMS, begging for handouts.
And RealityChuck, I had you until “hopefully.”