I'll answer this question first because I want to start more generally, and only get into specifics afterwards.
My thoughts in general are that I appreciate that both versions are here on BGA, and moreover that they are two distinct implementations, with different stats and Arena ladders and everything (even though I don't personally play Arena for this game). Having learned the classic and taught it to friends, I may eventually wish to try the newer edition, but I have no desire to see the old edition go.
I have some specific thoughts to your other questions, which I'll now address:
[*] Some beginners are lost because there are 2 different TTA on BGA, and are redirected to the "wrong" version (when you type "Through the Ages online" on Google for example, it directs you to the old edition). They may be disappointed which is sad for the TTA players community :(
If I were to search for "through the ages online", I would want exactly the result that you describe; if I wanted "through the ages a new story of civilization", then that's what I should be searching for. It's unfortunate that the designer chose a name so similar as to engender confusion (it's certainly made my life more difficult when searching for rule clarifications), but that's what we have to live with. :-p
To alleviate your concern, though, what about adding text on each of the game pages to point to the reciprocal game? Somewhere on the game page for the original you might have "This is the game page for the original classic Through the Ages. Looking for Through the Ages: A New Story of Civilization? Click here!"
Similarly, on the new edition's page, you could have "This is the game page for Through the Ages: A New Story of Civilization. Looking for the original classic Through the Ages? Click here!"
[*] The old TTA tables have difficulty to find opponents, while removing the old TTA will increase the number of TTA players by 2%, reducing the waiting time.
As for this, I would not assume that all players of TtA would automatically migrate to TtA: ANSoC.
[*] For the Arena: at now there are 2 "TTA Arena champions". The "new TTA champion" title is really something difficult to achieve, while the "old TTA" achievement is way more easy since there are 50 times less players...
I'm not sure why this is an issue. Games which are less popular are going to have fewer serious competitors than more popular games. I think that the people who know about the games will understand the differences and that comparing leagues would be an apples-to-oranges comparison. Presumably the league for the original TtA awards less XP for the champions and such, based on these lower participation rates: lesser rewards for an easier achievement—sounds reasonable to me.
In any event, thanks for soliciting player feedback!