Modules are "simpler" than vectors because they have fewer axioms, but they are also much harder to understand. For example, not all modules have a basis, which can make them much harder to work with.
Good luck explaining "simpler" with modules and vectors :).
Simple is defined as not to inter-wine. To understand an axiom is to understand how it "inter-wine" with other axioms to prove certain results. So fewer axioms necessarily results in more interwines, ie complex. I think here we are switching the subjects: from axiom itself to the results that we want to prove. If we focus on the simplicity of proving the results, the simplicity of axioms are irrelevant.
The main reason modules are interesting is not as a generalisation of vector spaces, but because they are helpful in studying rings. Kernels of ring homomorphisms are ideals, which in general are not subrings, but they are modules - and of course every ring is a module over itself. So to study a ring R it pays off to instead study R-modules, since working with them is... you guessed it! Simpler.
Simple is defined as not to inter-wine. To understand an axiom is to understand how it "inter-wine" with other axioms to prove certain results. So fewer axioms necessarily results in more interwines, ie complex. I think here we are switching the subjects: from axiom itself to the results that we want to prove. If we focus on the simplicity of proving the results, the simplicity of axioms are irrelevant.