How does podiatry foot surgery differ from orthopedic doctor foot surgery?

No difference. With training occuring side by side, podiatrists and foot and ankle ortho learn the same techniques. It is usually a personal preference on whom you want to see.
Same Principles. The principles are the same, it is a matter of selecting a surgeon with whom you have a good rapport and have confidence in. Look at the physician's education, training, and whether or not they are board certified. Also, take into consideration recommendations from friends and family that you trust.
None. It all comes down to how comfortable you feel with the Dr. The experience in performing the procedures is paramount. Not all procedures are done by all surgeons. Each surgeon has their own preferences as well.
Depends. A quality orthopedist and podiatrist are equal for foot surgery.Fellowship trained orthopedic foot and ankle surgeon is your best bet.
Specialization. Orthopedic surgeons can treat any part of the body, while podiatrist only treat the foot and ankle.
Not much. Foot surgery should be done by whichever is better trained and proficient at whatever procedure you are concerned with.