How are exponents and powers different?

An exponent is the number or variable that indicates the power to which a base number is raised. In the expression
2^4
[that's two with an exponent of 4--we use the ^ symbol around here because Answers won't let us write exponents as superscripts]
the 2 is the base number, the 4 is the exponent, and 2^4 (which equals 16) is the power of 2 that we're discussing.

There are two distinctions here. First, the exponent specifies which power; the power itself is the number resulting from performing the operation. (But that's getting technical, and we often seem to refer to the exponent as a "power" in expression like "two to the power of 4.")

Second, the exponent is a combination of a numeral and an operational indicator; it's a symbol indicating both the numerical power to which the base is raised, and the fact that the operation we're performing is raising that base to a power.

But the power itself is not a numeral or an operation; it's the resulting number. When we count by powers of 2, for example, we don't recite the exponents:
1, 2, 3, 4, 5 ...
Instead, we recite the powers themselves:
2, 4, 8, 16, 32 ...