Pro V1 or Pro V1x
As a relative beginner, I can't feel any difference between balls off the longer clubs. The only thing I've noticed is that the yellow range balls don't go as far as normal balls, but that's it . However, I can DEFINITELY feel a difference off the shorter clubs, wedges and putter, the putter being the main thing. I had a little experiment on my home putting mat, and with the half-a-dozen types of balls I have, there is a noticeable difference between all of them, mainly the pitch of the sound. Cheap balls seem to have a harder, high pitch, and almost dull "ping", whereas the better, more expensive balls have a much softer, lower pitch, and absorbing "thud".
Providing you don't spray them all over the place, or lose the plot when you have to hit over water hazards, I don't see any reason why anyone of any handicap shouldn't use a decent ball.