Both require and assert are used to perform certain checks during runtime to verify certain conditions.
So what is the basic difference bet
As Kigyo mentioned there is a semantic difference
There is also a major technical difference:
assert is annotated with @elidable(ASSERTION)
meaning you can compile your program with -Xelide-below ASSERTION or with -Xdisable-assertions and the compiler will not generate the bytecode for the assertions. This can significantly reduce bytecode size and improve performance if you have a large number of asserts.
Knowing this, you can use an assert to verify all the invariants everywhere in your program (all the preconditions/postconditions for every single method/function calls) and not pay the price in production.
You would usually have the "test" build with all the assertions enabled, it would be slower as it would verify all the assertions at all times, then you could have the "production" build of your product without the assertions, which you would eliminate all the internal state checks done through assertion
require is not elidable, it makes more sense for use in libraries (including internal libraries) to inform the caller of the preconditions to call a given method/function.