Kakashi with a Byakugan, I think the differences would be
He would not have 1000 jutsu, it would be more like 50 but they would be 20 times stronger.
He would have more stamina and chakra as the Byakugan would be less stressful on his body.
Raikiri as we know it would be changed. With Sharingan the Raikiri is a running high speed thrust, with the Sharingan giving you a counter to any counterattack. With the Byakugan an its ability to see the chakra points and channels, the Raikiri would be an up close high speed poke & slice(maybe just poke) combo that stabs charka points & cuts chakra channels.
Team 7 Vs Zabuza & Haku. Zabuza being a water ninja & Kakashi being lighting ninja would have an easier time beating Zabuza with stronger lighting jutsu an more stamina.
Kakashi Vs Itachi. It didn't matter if Kakashi had the Sharingan, Byakugan, Rinnegan or regular eyes, once he was caught in Tsukiyomi the match was over.
In the Chuunin exams Sasuke would still be able to learn the Chidora(this would be its first appearence) by appling the Sharingan to it. Kabuto would of had a harder time getting away.
Deidara Vs Kakashi. Diedara would be at a disadvantage to a more lighting jutsu & stamina based Kakashi, without Kamui it would take longer not just for Kakashi but Diedara too, as his lighting jutsu would diffuse most of his earth jutsu.
Pain Vs Kakashi. This fight Kakashi would still outclassed. Chouji probably would have die or Kakashi's death would be different maybe even both of them would be die...
Kakashi Vs Edo Zabuza. I see it play out basically the same, its still a bad match up for Edo Zabuza & Haku.