random.random() gives you a random floating point number in the range [0.0, 1.0) (so including 0.0, but not including 1.0 which is also known as a semi-open range). random.uniform(a, b) gives you a random floating point number in the range [a, b], (where rounding may end up giving you b).
The implementation of random.uniform() uses random.random() directly:
def uniform(self, a, b):
"Get a random number in the range [a, b) or [a, b] depending on rounding."
return a + (b-a) * self.random()
random.uniform(0, 1) is basically the same thing as random.random() (as 1.0 times float value closest to 1.0 still will give you float value closest to 1.0 there is no possibility of a rounding error there).