"Sometimes" merely indicates uncertainty. "At other times" would be a second statement, contrasting with the first. 1a) Sometimes the moon is full, and we can see clearly by moonlight. 1b) Sometimes the moon is full, and we can see clearly by moonlight. At other times, the moon may not even be visible in the sky, and there is no source of light but starlight. 2a) Sometimes we get poor reception from our TV antenna. 2b) Sometimes we get poor reception on our TV antenna; at other times, it is perfectly clear, just as good or better than cable. 3a) Sometimes migraine headaches are preceded by visual symptoms. 3b) Sometimes migraine headaches are preceded by visual symptoms; at other times, there is no warning at all.

It kind of means "sometimes", but there's a bit more nuance to it. It's really talking about different situations or circumstances. For example, you might say "I like going fishing. Every now and again I'll cast my line and get a bite pretty much right away. At other times, I'll be sitting on the riverbank for hours and hours before anything happens." So, "at other times" is used to introduce an alternative situation or case from the one you just described. That's different than saying "I like going fishing. Sometimes I get a bite right away" (which is only talking about one case). Or "Sometimes I sit for hours before I get a bite", which again, is talking about only one case.