You need both. If you don't have unit tests your feed back loop becomes way too slow, but then you of course have to test the "real" running of the program properly before shipping. And you need at least one person who didn't write the code to test it.
Unit tests are useful for some things. I can’t unit test my way to a game feeling good. For that I need a fast feedback loop to the real program. There’s more of that in most applications than most people suspect. And routes to get there. As programmers we make a lot of little decisions along the way and those don’t get proper consideration unless you can see that as the feature evolves.
