They are, for the most part, interchangeable. You could say, "from my point of view I'm not sure I quite agree" or, "from my standpoint I'm not sure I quite agree" and both are completely natural sentences. Standpoint perhaps implies a more deeply held, ideological belief. You might talk about a religious standpoint, or the standpoint of a socialist. Point of view doesn't imply quite the same conviction of belief. Your point of view is simply what you happen to think.