I do bike fitting and these are the things that I would think about, most others have already mentioned:

1) If the sole of the new shoe is thicker you will have to change the saddle height slightly upward as Oakleaf mentioned.
2) The cleat rotation may be off. Believe me, no matter how many bikes I fit, I still can reverse the rotation I want once in a while and it's really easy to shift the rotation slightly on one side as you tighten the cleat. Or maybe the cleat wasn't quite tight enough and it rotated a bit as you unclipped last.
3) See if where your foot is in the shoe is the same and/or the position of the cleat mount is the same. What I mean is, if in your old shoes X part of the cleat lines up with y part of your foot, make sure it's the same on the new shoes.

I hope that helps a bit.