Outside of mathematical logic, some familiar examples include:
compactness vs. sequential compactness—generalizing from metric to topological spaces
product topology vs. box topology—generalizing from finite to infinite product spaces
finite-dimensional vs. finitely generated (and related notions, e.g. finitely cogenerated)—generalizing from vector spaces to modules
pointwise convergence vs. uniform convergence vs. norm-convergence vs. convergence in the weak topology vs....—generalizing from sequences of numbers to sequences of functions
Outside of mathematical logic, some familiar examples include:
compactness vs. sequential compactness—generalizing from metric to topological spaces
product topology vs. box topology—generalizing from finite to infinite product spaces
finite-dimensional vs. finitely generated (and related notions, e.g. finitely cogenerated)—generalizing from vector spaces to modules
pointwise convergence vs. uniform convergence vs. norm-convergence vs. convergence in the weak topology vs....—generalizing from sequences of numbers to sequences of functions