Its called Royal (Dick) School of Veterinary Medicine because the founder of the vet school was William Dick. When I first heard about it I chuckled that I am going to be a Royal Dick.

There is also Dick Day where its basically like a powderpuff of the Glasgow vet school vs. Edinburgh vet school. I cant wait!! Go Edin!
I was in the same situation concerning whether I should apply to the UK as well as US schools. In the end, UK won out for me for various reasons:
1) I really liked that starting for Day 1 of your first year, you are working with animals. Unlike the majority of schools in the US, you do theory the 1st year or 2 THEN work with the animals during lecture.
2) Its AVMA accredited meaning my BVM&S (Bachelor of vet medicine and surgery) degree is the SAME as a DVM degree. I'll still be a licensed vet if I go back to the States. In fact, the BVM&S is licensed to the rest of Europe, (the states, obviously), South Africa, and Australia. Cha-ching!
3) I have dual citizenship with the US and Spain so I don't have to worry about student visas since I will be entering as an EU student.
4) If **** hits the fan in the US, you bet I will be permanently staying in Europe or vice versa. I am glad I have that option to grab my bags and go without having to deal with immigration laws.
5) I knew I wasn't going to get accepted with my instate school so going to Edinburgh comes out cheaper for me than if I went out of state.
6) UK schools don't care about GRE (thank god) and they actually look over your ENTIRE application before making a decision on your application. Unlike most schools they base the 1st trial of your application based off GPA and GRE score.
7) Yea its 3500 miles away, but the way I think of it, I'll still be away from my family, friends, and SO if I moved out of state across the country. The only drawback is that plane tix are much more expensive crossing the ocean but thats why you get Frequent Flyer Miles!
8) Finally, the next 5 years studying in Scotland are going to be the best years of my life and I can't wait to start!