Here are some differences
1. In Ultimate Team, you start out with a team of all crappy players, and gradually build your team by upgrading each position through packs and purchasing cards on the auction block. In CFM, you usually start with an average team and could get better or worse as time goes on.
2. Ultimate Team gives you the benefit of playing online head to head games and offline games against the CPU using the same team in the same mode. You can choose to only do one or the other if you want, but I think it is fun to mix the 2.
3. There is no salary cap or restrictions on building your team, and you can have virtually any player you want. For example, if you really like Jordy Nelson and Matt Forte, you could get these guys on your team relatively easily, where as in CFM, you need to give up significant pieces to get them.
4. Players don't age from season to season, so if you want Peyton Manning for 10 season in a row, you can use him.
5. I never spend money on the mode, and there is no better feeling than beating a guy with 99 OVRs at every position with your budget squad. I skipped Madden 25, but in Madden 13 I actually made several hundred dollars by selling MUT coins for real money.
6. If you want, you can build several different teams to play different styles of play.
7. You could say that opening packs is like having draft picks, and buying players off the auction is like signing free agents. I think opening packs is more exciting because you never know when you will get a rare card, and it is awesome when you do.
That is what I can think of off the top of my head