What happens if the house and the senate approve similar not identical bills?

Answer 1

Don't worry this will not happen. This is the reason why the Senate committees and House committees meet behind closed doors. These committees negotiate until they have created legislation that both committees find acceptable and will send to their respective legislative bodies. During this committee meeting the committees will agree upon language that will be sent to both the House and the Senate.

Answer 2

If there is any difference at all, even a single word, it goes to a Conference Committee made up of members of both the House and the Senate where compromises are made on the differences. The compromise bill is then returned to each house, with the same exact wording, for a final vote. Nothing is passed on to the President unless both houses have passed the same exact bill.