"They turned up at the party" and they "showed up at the party " both mean the same thing = they arrived at the party or they came to the party.

However, they both express more than that. It also gives a feeling that they arrived unexpectedly or after a long time.

Example:

"Why were you so upset last night?"

"My ex husband showed up at the party with his new wife!"

Another example:

"I haven't seen John in a long time"

"He turned up at the party last night!"

"No way! He didnt even tell me he was going!"

Also note "turn up" is slang for being wild and crazy like someone would be at a party

"We got so turned up last night at that party, it was a great time!

None really! They showed up, they turned up, they arrived, they made an appearance. Pretty much the same!