PhpSpec
PhpSpec is a design tool. It will make your life hard in the beginning if you're not used to TDD (test-first approach).
You'll learn a lot during the process though. You will also swear a lot. Many things, which are considered a bad practice, are not possible with PhpSpec (like partial mocks for example). You'll have to understand it and embrace it. Testing Laravel's facade's is hard, 'cause that's not OOP.
It's much harder to test an existing code with PhpSpec, since you have to make it testable first (which is easier with a test-first approach).
PhpUnit
On the other hand, PhpUnit is a testing tool. You can use it with TDD, BDD or whatever you call it. But you don't have to.
Choice
If your aim is to learn how to design better and you're commited to test-first approach, PhpSpec will help you. If your aim is to test or verify, PhpUnit is your friend.
TDD vs BDD
It's not either TDD or BDD. At least from a code level perspective, they're the same. BDD started as a TDD-redefined, offering a better terminology. Now it goes beyond the code level and it's so much more than that. These days I prefer to use a term proposed by Gojko Adzic - Specification by Example.