Bonus Question: Why are Royal Canadian Army Cadets 'Royal'?
Primarily because they follow British convention. The British Army is just that, unlike the Royal Navy and Royal Air Force. Although no longer true in usage, the difference comes from the fact that the British Army is/was technically loyal to the British Parliament, not the Monarch, as a result of the struggles between King and Parliament during the English Civil War. The navy was reinstituted after Charles II came to power and thus kept the 'Royal' title. As a difference its largely meaningless these days apart from its heritage elements. The same is true of the Australian Army, Royal Australian Navy, Royal Australian Air Force and their New Zealand equivalents.
/u/Thecna2 is correct on several fronts! The Canadian Armed Services derive their structure and heritage from the British Armed Services, and the Army of the United Kingdom does not have the prefix "Royal".
/u/Thecna2 is also correct in pinpointing the English Civil War as the progenitor of this tradition. The Army that won that struggle was the Parliamentarian Army, the "Roundheads", and when King Charles II was welcomed back on British soil it was this organisation that submitted to his authority and not the old Royalist Army.
This being so, there is a kind of "magic trick" going on behind the scenes. This Army had deposed his father, and Charlies II was not so keen to see that repeated; therefore Commonwealth Armies are technically "temporary" forces. Of course practically speaking, the British Army is a permanent force, but for a long time legally it was not. By divorcing the Army from the Crown that armed body was - perhaps still is - disbanded every five years and must be re-authorised by a law passed in Parliament. The Royal Navy and the Royal Air Force have never rebelled and deposed their Monarch; they are permanent organisations.
On the other hand, Regiments can have Royal in their titles. What's up with that? Well, Regiments are technically raised and wage war under the authority of their Colonel-in-Chief (generally today an honoury position) - not the Crown. This practice arises from the widespread use of private mercenary companies ("Colonel" was effectively the title for a mercenary company's CEO and often primary investor). Often these companies were named after their Colonel (for example, "Howard's Buffs" which later became the famous Royal East Kent Buffs Regiment). They were raised and commanded privately - often with Crown subsidies. If you have a regiment with "Royal" in front of it, this Colonel was/will be a member of the Commonwealth Royal Family. As Regiments are the building blocks which make up the army you therefore have the strange situation of some units being more "loyal" than others, with the result (once you lump them together) being institutionally mistrusted.
This is a very complex topic, with a fascinating answer; please ask questions if I was unclear. British military traditions often make no sense from an outside perspective.