The most important difference between linguistics and logic is that linguistics is descriptive while logic is normative.
Yes, some grammarians try to treat grammar as prescriptive, but most in modern times have given up on that and realize that usage will have its day and win out in the long run. And even when grammar appears to dictate form it does so only on the plane of signs, sans objects, and so remains a flat affair.
It is only logic that inhabits all three dimensions of sign relations, inquiring into how we ought to conduct our transactions with signs in order to realize their objectives. A normative science has different aims even when it looks on the same materials as a descriptive science. So logic may have abstractions from language among its materials but it is more than abstract linguistics — it is an augmentation of language.