According to ISTQB those two are not comparable. Functional testing is not integration testing.
Unit test is one of tests level and functional testing is type of testing.
Basically:
The function of a system (or component) is 'what it does'. This is typically described in a requirements specification, a functional specification, or in use cases.
while
Component testing, also known as unit, module and program testing, searches for defects in, and verifies the functioning of software (e.g. modules, programs, objects, classes, etc.) that are separately testable.
According to ISTQB component/unit test can be functional or not-functional:
Component testing may include testing of functionality and specific non-functional characteristics such as resource-behavior (e.g. memory leaks), performance or robustness testing, as well as structural testing (e.g. decision coverage).
Quotes from Foundations of software testing - ISTQB certification