Depends if there are any specifics you are looking for during your undergrad or afterwards. I think Edinburgh is slightly better known in the US but I could be wrong. The University of Edinburgh itself is very well known overseas because of its history. It also has an excellent informatics department: http://www.telegraph.co.uk/education...e-courses.html
If you're not that interested in research it won't make much of a difference to you. There are more module choices in Informatics at Edinburgh than Birmingham but not such a huge gap. If you prefer the city of Birmingham over Edinburgh and these factors of research aren't important to you, then I suggest you go to Birmingham since its CS program also looks quite good. Also, if you may be interested in student exchanges both universities have exchange program with ivy league universities (Bham: Cornell, Edin: Penn) so either way they are definitely well connected which is also an important factor.