Is there any difference between jump and jump up?
Bilbo jumped / Bilbo jumped up
"Up" adds direction to the description. If someone is seated, "jumped up" would refer to suddenly arising from the chair. Just "jumped" could mean that they were startled and jerked rather than arose.
Without a specified direction, it's more of a generic reference to leaping or moving suddenly and the direction would be surmised from the context. Skydivers jump, and people would surmise that they jump "out and down". A frog's jumping includes "up", but the important motion is forward.
In typical spoken language, people don't add a directional term to "jump" unless it is important to the context and the meaning would not be clear without it.
Suggests that Bilbo jumped more than once, maybe two or three times. If the statement were: “Bilbo danced”, it hints that he danced longer than a few seconds.
Bilbo jumped up and down
This statement tells us that Bilbo jumped repeatedly. If we say that “Bilbo danced round and round” it suggests that he danced around the dance floor more than once, possibly many times.
Bilbo jumped up
This declarative sentence seems to suggest that Bilbo was either sitting or in a stationary position when he suddenly jumped up from his place.
Only the context will clarify meaning.