The UK has a Royal Air Force and a Royal Navy, even a Royal Lifesaving Service IIRC, but the Army is just the British Army. I believe some individual regiments have their royal warrants, but why doesn’t the Army have the same status as the Air Force and Navy? Is this just some historical oversight that has hardened into a point of pride?
The Army is a collection of branches and units, most of which have their own “Royal” label:
Royal Army Medical Corps (RAMC)
Royal Regiment of Artillery
Royal Armoured Corps (RAC)
Royal Corps of Signals
Corps of Royal Engineers
Royal Logistics Corps
Corps of Royal Electrical and Mechanical Engineers (REME)
Royal Army Chaplains
Infantry units tend to have names related to specific traditions, ceremonies or royals (e.g. the King’s Own Scottish Borderers, the Prince of Wales’ Own Regiment of Yorkshire, Queen’s Royal Hussars, Royal Gurkha Rifles
(And don’t forget the Royal Marines!)
Here’s some wild speculation: the army is an older structure than the navy (and, obviously, the air force). Back in olden times, the nobles would have separate armies, which eventually consolidated into the British Army. The navy, on the other hand, was a more centralized (and expensive) structure under royal control, hence, the Royal Navy. I have no idea if this is at all accurate, but I like the way it sounds.
My favorite infantry unit name is the Black Watch.
My understanding, as a lawyer but on the wrong side of the ocean, 
is that the difference is that the existence of, and appropriations for, the army have to be approved each and every year by Parliament while the navy and air force are ongoing institutions and can have long-term appropriations.
I would be surprised if the above rule and the British Army/Royal Navy distinction didn’t stem from the English Civil War. It has to do with the fear of standing armies, which can be used for domestic repression, versus a navy which is generally useful only against external enemies. Or, to put it another way, an island nation needs a permanent navy but has little need for a standing army, so the king or queen better get Parliament’s permission to have one to make sure he/she doesn’t want one for the wrong reasons.
Certainly, the “need a navy/don’t need an army because we’re isolated by water” logic manifested itself in the U.S. Constitution, where a two-year limit is placed on appropriations for the army but no such limit is placed upon the creation and funding of a navy.
IIRC the Royal Navy and The Royal Air Force have to be funded by vote annually the same as the Army.
I think that JeffB is most likely to be right with the historic founding of the Armies, but the Navy and Airforce being more regularly founded.
It is probably too much to expect a simple answer to this question. This is just the way it is. There is no obvious reason why the names should be consistent.
The reason I’m suspicious of all the above explanations is that the term Royal Navy is a rather late coinage. My impression is that it does not become widespread until the eighteenth century and then only as an unofficial name. By then both services were raised, organised and financed along roughly similar lines. It is true, as John Bredin points out, that there was a stigma attached to the idea of a standing army, but everything about the army’s uniforms, traditions and regimental names celebrated the fact that this was the king’s army. The memory of the Civil War made it unthinkable that it could be Parliament’s army.
I suspect that the real answer is mere linguistic fashion. Both ‘the Royal Navy’ and ‘the British Army’ are cliches. They are also ever so slightly easier to say than either ‘the British Navy’ or ‘the Royal Army’. In the case of ‘the Royal Navy’, its use is so engrained that it has become that organisation’s official name. I don’t think it was ever formally granted.
The RAF is a special case as, when the name was granted, it was a subsidiary organisation in the way that the Royal Army Medical Corp etc. still are. The idea of a link with the Royal Family is all the more sought-after for those units which are small, new and unglamourous.