Depends whether or not Tenshinhan joins in when Piccolo goes to train with Goku and co. If not, he doesn't make any new major contributions to the plot and at best manages to retain the title of strongest Earthling. If he goes to train with them, he'd still probably end up lagging behind the Base Saiyans a little when considering Gohan didn't surpass the other 3, though everyone having knowledge of the Shin Kikoho and likely having better opportunities to enhance it would allow them to formulate strategies to use it tactically. For the sake of the scenario, we'll say Piccolo plans for Kuririn to use a Taiyoken once Vegeta goes down against #18, have Trunks get him out of the way and have Tenshinhan fire off a Shin Kikoho. The plan is a success and, through that alone or a combined attack backing him up, Tenshinhan destroys the twins. #16 isn't hit and flies off to Goku's house, nobody being able to combat him. Piccolo immediately goes to fuse with Kami but nobody is able to stop #16 from killing Goku before he can be transported elsewhere. Piccolo still confronts Cell and things go as usual, though the power of #16 convinces him the Rosat may be essential to combat both threats. He goes in with Tenshinhan and, even should Cell gain as much power as in canon, he's easily killed by Piccolo once he can be found as would #16 be eventually through alternative means of searching for him (Looking from the Heavenly Realm, asking Baba, etc.). Piccolo would then converse with Kaio to find the location of New Namek, Bulma would build a ship based on her father's previous design and they'd revive Goku and acquire Dende as the new Kami. Vegeta and Trunks would also go in the Rosat and Trunks would return to his future afterwards. Goku would also go in the Rosat with Gohan on Vegeta's request to get in shape for their rematch quickly, though Goku's power upon emerging would shut that idea down.
7 years later, both Goku and Vegeta would have achieved SS2, so they're able to take down Dabura either alone or together even with Vegeta being slightly weaker than his canon self regardless of where you rank Dabura compared to other characters. Goku not having Otherworld training means Vegeta isn't desperate enough to get possessed and Babidi's plans are foiled. Goku and Vegeta have their rematch, which likely ends in a draw.
As for Super, it plays out near enough the same beyond Beerus not threatening to destroy Earth and instead asking the Saiyans cordially about SSG after he's had some pudding. This is until the Future Trunks Arc which wouldn't happen at all due to the lack of the 2nd Rosat trip leaving Trunks far too underprepared for Dabura and being killed. Zamasu's Zero Mortal Plan goes undetected for a while until it becomes too blatant for Daishinkan or Zen-Oh to not notice. Goku not achieving PSSB here nor Vegeta figuring out it was possible means things go one of two ways; either they still achieve it on their own and things remain unchanged (minus the lack of his potential being unleashed meaning Gohan jobs to Kefla and some like Jiren has to deal with her) or Goku and Vegeta are beaten by Jiren and Toppo quickly and Universe 11 wins the ToP.