I agree with the answers by @Alex Fleischer and @Marco Lübbecke as essays in their own right, divorced from the question which was asked. However, neither of them directly address the question as asked, which is specifically about MINLP - Mixed-Integer Nonlinear Program, and not about LP or MILP.
MINLP solvers (not counting convex conic MINLP solvers) generally employ derivatives (gradient and maybe Hessian of Lagrangian, Jacobian and maybe Hessians of constraints). Solvers may have various derivative options (forward or central finite difference), "exact" (analytical or automatic differentiation). Different modeling systems may provide different derivatives to the solver - some only utilize whatever is provided by the user, and may default to finite difference if none are provided; whereas others, such as AMPL, automatically compute and provide automatic derivatives to the solver, and yet others, such as YALMIP, automatically provide first derivatives (gradient and Jacobian), but no 2nd derivatives, and no option to provide 2nd derivatives. The derivative options (leaving aside the possibility of coding errors) can have a large impact on (MI)NLP performance (for example, finite difference Quasi-Newton vs. Quasi-Newton vs. Newton for continuous relaxations). Note that some MINLP solvers, such as BARON, automatically compute derivatives internally, in which case modeling system derivative options are irrelevant to solver performance.
Also, modeling systems may differ in the type and amount, if any, of pre-solve performed by the modeling system prior to providing a problem to the solver. The pre-solve performed by the modeling system can have a large impact on MINLP solver performance, and may not be the same as the pre-solve which would have been performed by the solver.
Also, note that there may be different default starting (initial) values provided for the variables, depending on the modeling system (and its pre-solve). Generally, a user is allowed to provide starting values; but if not, the modeling system default can have a big impact (do you know how often the all zero vector is provided as default starting value, and there is either a model singularity there or non-local optimum stationary point?).
Default values of other solver algorithm and parameter choices may also differ by modeling system, and have a large impact on performance.