How can I differ feature testing from functionality testing?

Unlike for other Testing types, for the term Feature testing, you can find only a best match from well accepted testing types in Software QA domain.

Feature testing - I think it is the sanity test. In a mature build, particular feature could be newly added, or existing feature's bug could be fixed. Then the testing has to be carried out focusing the particular feature alone. The sanity test of a mature build is also the same thing. However feature test (sanity test) alone would not qualify application for sending live. The regression test has to follow right after to make sure nothing in the whole system is broken by new addition of feature or bug fix. The link here should give you more understanding on Sanity test Functional Testing has 2 interesting meanings depending on the context of testing

As a sub type of System Testing Types

Functional Testing - Testing to look for missing functionalities which could be useful to the end users

This is not what the simple name suggest what it could be. It is one variation out of 7 main types of system testing. System testing is also called End to end testing. One major aspect of system testing is Functional testing where testers look for missing functionalities which could be useful to the end users. Idea is to improve the quality of application by new functionality suggestions after the test. Please refer this link for more info on system testing Main testing type that cover many other sub testing types Functional Testing - Test how system functions conform to the functional requirements provided by the customer. Composed of many testing sub types involving even developers in the team to contribute. Check this link for more information on this main test type.