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