Parametric test:

It will determine specific parameters such as current & voltage levels.

Parametric tests are undertaken to measure specific electrical characteristics of the device.

DC parametric tests include contact test that ensures that there is no open or shorts on chip pin, it measures worst case static and dynamic power dissipation, it performs output short current test that measures the output current at low and high output voltages.

It performs threshold voltage test that measures $V_{IL}$ and $V_{IH}$ values. These are not time dependent.

AC parameter tests measure time dependent characteristics such as delays rise time, fall times of signal.

It also includes setup & hold test.

Functional test:

It will exercise the design in such a way as to exercise the operation of the design through the various functional operations that it would be designed to undertake.

For complex digital circuits & systems this can be extremely time consuming & costly.

It may be applied at elevated temp to guarantee specifications.