AE's are great budget dress shoes. I also don't mean "budget" as in poor quality, I mean it as - the best possible shoe at their price point. Also check out their factory shoe bank online - it is a great way to purchase very inexpensive shoes with minor imperfections.
To your question - functionality of the two shoes are identical. Most AEs use 360 goodyear welting. Balmorals (or Oxfords) are much more formal. To really answer your question, you can wear whatever the hell you want. You are your own man. But, as a basic rule, the differences come down to more or less formal appearances.
For example, if you wear a full suit everyday, i would first purchase black balmorals. If you wear khakis/slacks and a blazer or sportcoat, I would first get black cap-toe bluchers.
The firm I work for is pretty laid back. For example, it is Friday, and I am slacking off a bit in jeans and a sweater writing this post... I typically wear slacks, a nice shirt/tie, and no jacket. So I wear black bluchers with my charcoal wool trousers. I also wear my wedding shoes (gorgeous Crockett & Jones walnut balmorals) fairly frequently - with navy slacks, light grey slacks, even nice khakis and a sport coat. Others may disagree, but the lighter brown, more intense burgundy/oxblood colors are less formal to me. So IMO, you dont look tacky wearing those brighter colors on a balmoral shoe - even if you are not in a full suit.
To be honest, I dont think it matters (as in, really really matters) unless we are talking about black shoes w/ suits or tuxedos. Don't get me started on black patent... The more important thing, I think, is the last matching your foot. A blucher that fits you like a tight glove will look sleeker and feel better than a clunky balmoral. Going to an AE, Alden, or any other nice shoe store and experimenting with the lasts will pay huge dividends later on - trust me. Nothing is worse than dropping serious coin on a shoe that still doesnt feel right after 3 months. If you select the right shoe/correct last, it will feel better than your sneakers/running shoes in 3-4 months - I guarantee it.
Cherrs my friend