Run around obviously has a literal meaning, you can run around the track, but otherwise run around and about mean to just be busy, go different places, do different things, mostly chores or little tasks you have around or just generally be busy.
跑來跑去顯然有字面意思,你可以在跑道上跑來跑去,但除此之外,跑來跑去意味著只是忙碌,去不同的地方,做不同的事情,主要是你身邊的雜事或小任務,或者只是一般的忙碌。