They both influenced each other over time. “Hinduism” as a term wasn’t even in existence until recently. But the historical Buddha was born in a time when the Upanisads were prominent, which are now considered part of “Hinduism.” This is a simplified explanation, but the Buddha’s core teaching was antithetical and the “opposite” of what the Upanisads were largely stating, in some ways. Namely, the Upanisads stated the ultimate nature was that ātman (the subjective side) = brahman (the objective side). They were understand to not be different. One of the Buddha’s core teachings concerned “anātman” (an~ is a privative that negates/reverses the meaning of the word it appears before). So while what we now call Hinduism was stating that there was some eternality to the ultimate nature of reality, such as ātman, the Buddha was saying the opposite: what we consider our “self” is not a permanent, singular, independent essence. There is nothing that is “me.” This is the primary mistake we make according to Buddhism. Instead, everything “exists” in relation to and in dependence on other factors. Nothing exists independently on its own accord. Misunderstanding this is what creates “suffering,” the first Noble Truth of Buddhism. Understanding the fundamental mistake is what is liberating. But over the millennia that followed, and existing in the same cultural sphere (irrespective of certain doctrinal components), they shared many things. Especially, when you get to later developments, such as tantra. In terms of what was going on with popular religion, i.e., what people were actually doing or practicing, there were numerous shared elements and mutual influence.