There isn't much real difference, no. Perhaps "tidy up/clean up" involves doing a more thorough job including discarding things, as opposed to just "tidy" which might imply merely making things look neat.
There is not a big difference but adding "up" conveys a sense of more complete. Tidy means to make somewhat tidier, while "tidy up" implies completing the job of making the room tidy. One of the meanings of "up" in the WRF dictionary
(particle) indicating intensity or completion of an action: he tore up the cheque, drink up now!
(1) "I'm going to tidy up the room."
(2) "I'm going to tidy up." (Direct object, such as "the room", is understood.)
(3) "I'm going to tidy the room." (Less frequent than (1).)
(4) "I'm going to tidy."
I was searching for the difference between 'tidy' and 'tidy up' and found this old post. I understand the meaning in above posts. But I wonder if this would vary by the items we're dealing with. For example, if it's a 'toy' not a 'room' should we say:
'1) Kids should tidy their toys.' Or '2) Kids should tidy up their toys.'?
I've never heard "toys" as the object of tidy or tidy up. To me, tidy and tidy up both have to do with the general appearance of a room or space. It can't be applied to specific objects. I can tidy up my desk but I can't tidy up my pencils, for example.
The verb "tidy" sounds odd to me without "up".
The same "up" is used with a number of verbs to mean "to completion".
To clean something up, to write up your notes, to grow up (grow to completion), to build up, to break up,...