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When you write:

sage: math.exp(2)
7.38905609893065

you rely on the exp function provided by Python math numerical library.

When you write:

sage: exp(2) e^2

you rely on the Sage exp function, which by default returns a symbolic expressions. Tou can get its numerical value (with arbitrary precision) as:

sage: exp(2).n()
7.38905609893065
sage: exp(2).n(digits=100)
7.389056098930650227230427460575007813180315570551847324087127822522573796079057763384312485079121795

When you write:

sage: math.exp(2)
7.38905609893065

you rely on the exp function provided by Python math numerical library.

When you write:

sage: exp(2)
e^2
e^2

sage: exp(2).n()
7.38905609893065
sage: exp(2).n(digits=100)
7.389056098930650227230427460575007813180315570551847324087127822522573796079057763384312485079121795