If the dirty dishes are on the table in the dining room, the sequence (in my use of the verbs) is:
- Clear the table / Clear the dishes away [the dirty dishes are not ready to be put away - they have just been moved to the kitchen]
- Wash up
- Dry up
- Put the dishes away
The dictionary that told you the two phrasal verbs 'had the same meaning' was over-simplifying (probably for lack of space). There are contexts where their meanings are similar.