In that case, neither of those cpus are going to show a noticeable increase in performance, unless you have apps or games(very few at the moment) that will take advantage of the extra cores/threads. I'd say go for the monitor next, but be aware that a 1080 isn't the best at running 4k. You will need to turn settings down in some games.
There could be any number of issues with the stutter. What's in the rest of your current system?
It could be drivers, v-sync, etc... would need to narrow it down.