When you have a model structure, the cofibrations and fibrations give you the acyclic cofibrations and acyclic fibrations because of the lifting axioms. Then, the weak equivalences are those arrows which can be factored as an acyclic cofibration followed by an acyclic fibration. (Use the factorization and $2$ out of $3$ axioms.) That is how the class of cofibrations and the class of fibrations give you the whole model structure.

When you have a model structure, the cofibrations and fibrations give you the acyclic cofibrations and acyclic fibrations because of the lifting axioms. Then, the weak equivalences are those arrows which can be factored as an acyclic cofibration followed by an acyclic fibration. (Use the factorization and $2$ out of $3$ axioms.) That is how the class of cofibrations and the class of fibrations give you the whole model structure.

EDIT: While I am at it, it may be worth mentioning that you do not need all the fibrations to recover the model structure once you know the cofibrations. The fibrations whose codomain is the terminal object give you a sufficient data. In other words: a model structure is determined by the cofibrations and fibrant objects. I think this observation is due to Joyal. This is Proposition E.1.10. of his text The Theory of Quasi-Categories and its Applications.