There are some differences. For example Bishop-Phelps theorem, which holds only in real Banach spaces. On my opinion, this qualifies as a "major theorem".

MR1749671  
Lomonosov, Victor
A counterexample to the Bishop-Phelps theorem in complex spaces.  
Israel J. Math. 115 (2000), 25–28. 

Remark. Your statement "natural to assume that the field is real if the problem comes from physics" is completely wrong. 

In fact physicists are MORE interested in the complex filed than in the real field. The most fundamental theory of physics, quantum mechanics, describes the state of a system as a vector in the COMPLEX Hilbert space. From the point of view of physics, real numbers are just eigenvalues of Hermitean operators on a complex Hilbert space.