The ‘Live trains’ tab shows direct trains departing or arriving soon. It’s equivalent to the information boards you see in stations. So it’s great for up-to-date and full information.
The Planner tab features a full journey planner, which can plan journeys in the future and also sells tickets. So it’s great for multi-part journeys and for buying tickets.

The reason there are two different tabs is that each uses a different information source.