Is there a difference between mathematical logic … and… logic?

Answer

In a way.

Mathematical logic is logic used in Math.
Logic is the arguement consisting of one or more premis, deduction and/or supportive statements.

The structure in its most primitive form is :
Premise : Start statement.
Arguement : A subject to which our premise is applied to.
Conclusion : “logical deduction”.

The idea for all logic is that it depends on unanimously accepting the truth of the premise. In other words the truth of the logic (conclusion ) is dependent on the “premise”, crudely speaking. Hence the limit of logic because one can always refute a “logic” by nullifying the “belief” on its premise.

We can go on “logically” proving every premise since there either will come a “super premise” or “premises go infinitely”. The former is the case while the second is an absurdity. Thus this “super premise” must be accepted without any “logic” to back its validity.