A tensor category includes the information of a tensor product, which is something that takes objects and returns objects. This means that a tensor functor can't just "preserve tensor product" it needs to have higher data which gives isomorphisms between $F(X \otimes Y) \rightarrow F(X) \otimes F(Y)$ satisfying some naturality conditions.
But braided and ribbon structures are things that take objects and return morphisms. So they're up one dimension higher. This makes life easier. For example a braiding is an iso $\sigma_{X,Y}: X\otimes Y \rightarrow Y \otimes X$ and a monoidal functor is braided if $\mathcal{F}(\sigma_{X,Y})=\sigma_{\mathcal{F}(X),\mathcal{F}(Y)}$. This is just a condition, not a structure. Similarly, a spherical structure (which is equivalent to a ribbon structure) assigns to every object a morphism $f_X: X \rightarrow X^{**}$, and a functor preserves ribbon structure if $\mathcal{F}(f_X) = f_{\mathcal{F}(X)}$. Again this is just a condition not something complicated.
Modularity is just a condition and not a structure at all. This means that there's just not difference between a ribbon equivalence and a modular equivalence. If two ribbon categories are equivalent as ribbon categories and one ways modular then the other must be modular too.
