You can expect the Byakugan to give the user more insight on things like jutsu than the Sharingan. An example would probably include Neji identifying Kidomaro's build up of chakra before an attack. Its near 360 degree and penetrating vision (along with telescopic abilities) gives the Hyuuga clan a great defense in battle as well as excellent sensory abilities. Correct me if I'm wrong but couldn't a Hyuuga see through the hidden mist? Personally, I've seen flaws with this vision in some parts of the manga (not regarding the blind spot), but overall it's pretty impressive. IIRC the Byakugan is technically not the source of gentle fist or rotation, but it builds the foundation for it.
The Sharingan is less potent in terms of sight. However it does offer some perks over the Sharingan. Its copying abilities are the main reason Kakashi got his nick-name as the "copy-cat ninja" and the hype of copying 1,000 jutsu. You can copy taijutsu, speed, ninjutsu, genjutsu (I think), and even hand-writing if you have the abilities for them. Other benefits includes precognition. The boost from 2-tomoe prediction to 3-tomoe was the difference between Sasuke getting blitz'd and being able to blitz without getting hit once. Furthermore, it allows potent genjutsu, which Itachi shows handily. I don't THINK I'm forgetting anything else...
It depends on whose using them, but I guess you could also compare weapon vs. weapon if need be.