I own both rule sets for DH and BA:
DH was my first rules set for WW2 gaming. It is clearly more historical and detailed in complex elements in many respects. It is a good system, by my experience also takes longer to play and can be a bit more confusing for a new player until you really learn the rules, but you do get more detail to the game if you are into that realistic ww2 feel.
BA is more simplified, perhaps to the experienced ww2 veteran gamer, a bit too much. The games to me play smoother, faster and yet covers enough of the rules to keep the game balanced and fair. The system is very unique in the blind draw of action dice of units that makes the game both unique and dynamic. The use of the pin rules also is another unique element I like, it reflects psychology of combat in a simple to understand way to again make it easy for newer players to catch on. BA is clearly more newbie friendly and is a bit more "all ages" compared to DH which does require a more mature and veteran wargamer that has more time on their hands to play a game and be willing to deal with more complex elements. The all ages aspect also means that younger players can be taught the finer elements of other historical games that are not pop culture like War Machine/Hordes or Warhammer that there are other game systems to explore and enjoy. Besides once all of us old fockers die off who is going to play these games?

BA has been well supported which is about the only complaint I would have about DH is lack of add on materials....