I admit, if you showed me these positions cold and asked me which black should prefer, I would have said the first. But given your introduction, let's look for things that favor black in the second diagram.
The first thing that I notice is space. The pawn structures of the two diagrams are the same, but since black has less space, trading a pair of minor pieces makes development easier. In the first diagram, black has two minor pieces whose only safe development square is e7. They can't both occupy that square so black has a space issue. In the second diagram, the abscence of the g8 knight alleviates that space issue.
In the second diagram white has traded the c1 bishop. This is technically a bad bishop, but could serve two very useful purposes in this game. The first is guarding b2. Since black has the powerful queen move Qb6 double attacking b2 and d4, it's nice to have a defender of b2 and there is none in diagram 2. Also, even though this bishop is bad, it might easily participate in a kingside attack if black castles kingside (Qg4, Bh6).
That's all I got. :-)
Position #1 is the French Defense / Advance Variation where White scores 51.2%. Position #2 is a variation in the Trompowsky where White scores 41.7% and several GMs analyzing this position claim White has been "refuted". White to move in both cases.
Why is the second position worse for White than the first position?
In the second position, White's traded off his bad bishop (in the French Advance, trying to get rid of your bad bishop is a typical theme, so "losing the bishop pair" should at least not be a disadvantage in this case), Black's played h6 instead of Nc6 (and h6 does less to aid Black's c5 push), and Black's one tempo down. None of these factors seem to favor Black, and yet Black tends to score more often in the second position.