I would say that the bow is great at all ranges and (compared to the HR) is quite good in mid-range and close combat (especially when upgraded).
If I have an enemy rushing me (mostly with an upgraded bow), I can get the down a decent amount of the time (say 65-70% of the time)... Much like the HR in that situation, I find it comes down to reaction time... in such cases, I often do not try to get the headshot but rather the 2 or 3 required body shots. Once upgraded, the bow can fire at a pretty decent speed (arguably not usually fast enough to beat out a flamethrower most of the time)
At long range, it is a bit harder... Like ACP said, you know you are going to hit the target (if they hold still for a second) when the dot is a closed circle (which can also be helpful if there is smoke or weather obstructions)... As the weapon does not have a scope, it is a bit harder to get headshots, but it can be done... ex. I was on the bridge of the sam the other day... and enemy was in the water behind a rock... the only visible part was their head... the circle turned solid... headshot (which, as a few people said, is quite satisfying). I find this just takes some practice once you have a feel for the bow's mechanics. One good thing is that the bow, when first drawn, doesn't sway too much so you can more easily line up the shot (compared to SP).
As foofighter mentioned stages, I would agree that Downtown is generally the one map that makes the bow hard to use... mostly because of the many winding turns and obstructions... plus... because the board is dark and the bow is a weapon of accuracy, it is often hard to even see the enemy let alone aim. The other boards are pretty well suited for the bow.
Stalkers- Covert Training 3 users
Clickers- Constant reloaders
Bloaters- People getting blown up by bombs