I googled about this and the descriptions for both types of testing seem to be very similar but I am not sure if they are both the same.
3 Answers
"Feature testing" and "Regression testing" are two slightly different terms.
Feature testing is a testing of some functionality, which is typically limited within some logical area. For example you want to test whether the "Like" button on Facebook works properly. The testing activities related to that "Like" button can be named "Feature testing". I would also treat the word "feature" in "feature testing" term as just a piece of functionality.
Regression testing is a process of making sure that changes into existing database didn't cause new defects. In case of regression testing of "Like" button on Facebook, you would test that after each update of button's functionality is still works as expected.
I would also say that term "Feature testing" doesn't belong to the terms which are used "officially", since "feature" can be everything and therefore it has too broad meaning.