Cut open a watermelon and you'll see plenty of seeds — seedless or not. They may be black or white, but they're seeds just the same. Is there a difference between the two colors? Do you have to spit out both?
The Short Answer
Yes, they are different, and no, you don't have to spit out both.
The black seeds in a regular watermelon are just plain ol' seeds. They're mature, fertile seeds, so if you planted a few in the ground, they would indeed sprout into watermelon plants. While they're thought to be too hard and therefore inedible, they are actually completely safe to consume.
White seeds are really just immature black seeds. They're soft and almost translucent because they are only just beginning to grow and develop. The seeds are infertile, so they won't sprout into a watermelon plant if you planted them. They are, however, so soft that they aren't a bother to swallow along with the flesh. While you'll find white seeds in regular, seeded watermelons, you'll find even more in seedless watermelons, which were made to not produce mature seeds, so all the seeds in the fruit remain immature and white.