They have similar meanings, slightly different usages... Sunrise can also be a noun. For example. We watched the sunrise. (sunrise=noun) Focus on the thing, the sunrise -- it was so beautiful, the colours were amazing... We watched the sun rise. (sun=noun, rise = verb) Focus on both, the thing and the action -- maybe if you're there with a boyfriend, you woke up from sleeping on the beach, you were still there at 5AM, sky was light and then you saw the sun rise... We watched the sun come up is more like the second example, but this is hair-splitting meaning almost to the point of not being useful ... everyone would have a different interpretation and none would be wrong...

They mean the same thing, in general. Just depends on the context. I'd use "we watched the sun come up" if I had waited the whole night for that, and "we watched the sun rise" if I had only seen dawn.