BOTH. Both aspects are important - everywhere on the course. I'm thinking more specifically as it pertains to putting. It's a question I field as often as any other from my students. Should I be more worried about the direction of the putt of the distance control?
I'm going to say both are important. I'm not trying to get out of answering the question, and I'll explain what I mean in minute, but both distance AND direction are key to making more putts. I need greater context to give you my opinion on which is the more important key depending on what the putt looks like.
Unfortunately, the only real data I have is from PGA Tour stats - not from average golfers. That said, I'll do my best to translate or equate these to what you, me and "Joe Shmoe" should expect on the greens. Now these statistics are coming from GolfTec and are based on tour averages for 2016. Here we go:
At approximately 35 feet, touring pros average more '3-putts' than they do '1-putts'. Though the percentages are very close, 35 feet seems to the breaking point. Far and away, 2 putt performances outweigh 1-putts and 3-putts on the greens (outside of 8 feet). I think we can say with a fair bit of confidence that the distance from which an average golfer is more likely to hit 3 putts than they are 1 putt, is MUCH closer to the hole (let's say about half the distance of the tour guys). So from 16-20 feet, average joe golfers are more likely to 3-putt than they are to make the putt. Sounds accurate to me!
This is the context I needed to respond to the question, "distance or direction?":
Are you 6 feet from the hole or are you 30 feet from the hole?
I'm much more firmly planted in the "distance" camp than I am in the "direction" camp when it comes to what's more important (overall). But I will argue that, based on the stats shown above, if you're inside that 16-20ft range (better players move back a couple of feet and those less apt for success on the greens, move closer) focus on your line. If you are beyond that 16-20ft range, focus on the speed. In my teaching experience, the issue with most golfers isn't so much about how to line up their putter but more about CHOOSING a line on which to start the ball rolling (reading greens). I'm saying that on a straight putt, the average joe does a pretty good job of getting the ball rolling in the general direction of the hole (typically not missing by more than a few inches on either side). The issue I see more frequently, is blowing the putt 6 feet past or leaving it short (a cardinal sin)! No matter the direction, you are going to miss every single putt that you leave short of the hole. When you're closer to the hole, your putting stroke requires less motion - meaning it's much less likely you'll struggle with your distance control. When you're further from the hole, your putting stroke requires a larger motion - and that's when our distance control will suffer.
All that said, know the context. If you're 30 feet from the hole, get your line and be confident in your reading of the green, but focus more attention on the speed needed. If you're 6 feet from the hole, make an appropriate length stroke, but make sure you're committed to your line!