Seeing someone usually applies to the beginning of a new relationship. It usually indicates a heightened level of interest, and even offers subtle speculation that it may turn into a serious, committed relationship. Dating tends to imply that the relationship has grown into a much more serious endeavor.

What does I am seeing someone mean?

People who are seeing someone are dating. They go out on dates regularly with the person that they are seeing. The whole point of seeing someone is that the person sees the other person often, usually at least twice a week, if not more.