The view of the whole is greater than the sum of its parts. And that includes tests for the whole compared to tests of single things.
First, I just don't want to into a discussion about what's the "unit in a unit test"1, so let's take the point that a unit test is a test that tests a class/function, not the whole system from end to end, which would require data flowing through several classes/functions.
There are several libraries/frameworks that actually split this in a way that you can't test the whole. Spring+Mockito is one of those combinations -- and one that I worked with. Due the bean container of Java, the extensive use of Beans by Spring and the way Mockito interacts with the container, it's pretty easy to write tests that involve only one class: You can ask Mockito to mock every dependency injection (so it injects mocked beans instead of the real ones) in one class and mock every injected class, simply using annotations.
And this is cool and all and makes tests simple and fast. But the fact that we are making sure each class does what it should do, it doesn't give a proper view of the whole; you can't see if that collection of perfectly tested classes actually solve the problem the system is responsible for solving.
Once, in C++, I wrote an alarm system daemon for switches. There were three different levels of things the alarm system should do, depending on the incoming message from a service: It could only log the message of the incoming error, it could log the error and send a SNMP message, or it could log the error, send a SNMP message and turn a LED in the front panel on. Because each piece had a well defined functionality, we broke the system in three different parts: One for the log, one for the SNMP and one for the LED. All tested, all pretty. But I still had a nagging feeling that something was missing. That's when I wrote a test that would bring the daemon up, send some alarms and see the results.
And, although each module was well tested, we still got one things we were doing it wrong. If we never wrote an integration test, we would never catch those.
Not only that, but because we wrote a test that interacted with the daemon, we could get a better picture of its functionality and the test actually made sense -- as in, if you read the unit tests, they seemed disconnected from what the daemon was expected to do, but the integration tests actually read like "Here, let me show that we actually did what you asked". And yes, this was akin to Gherkin tests, although I didn't know Gherkin at the time -- and, better yet, we had tests that proved that we were following the spec.
Personally, I think over time integration tests become more important than unit tests. The reason is that I personally have the feeling2 that unit tests check if the classes/functions have adherence to the underlying design -- Does your view can actually work without the controller? Is the controller using something from the model or using things that should be in the view? -- but adherence to the design gets better over time -- developers start using the layout from previous examples, so they capture the design by osmosis, while the big picture starts to get more and more complex, with lots of moving parts.
There is no "unit" in "unit tests". "Unit test" means the test is a unit, indivisible and dependent only on itself.
Again, it's pure feeling from my experience. I have no data to back that affirmation up, so take it with a grain of salt.