Binary lifting: preprocessing, queries. Simple to implement and extend to weighted paths. Euler tour + RMQ: preprocessing, queries. Faster for large , but more complex and uses more space. For most problems, binary lifting is enough. The implementation is cleaner and easier to debug.
Use Euler tour + RMQ when or when you need the absolute fastest queries. In competitive programming, binary lifting is the default choice. Only switch to Euler tour if you are certain the time limit requires queries. The added complexity is not worth it for typical constraints.