There are a few answers discussing the relative merits of rifles and bows, and we all know that rifles are going to be the better general-purpose tool. That's not really news. That's probably not what you're after, though. The basic principle of armed conflict hasn't changed since the bronze age: stick them with the pointy end. Arrows do that just fine. The question then becomes, why can't I put a cool guy in a story with a compound bow and a thick quiver?
Plastic-based and ceramic-based armor is basically impervious to arrow strikes. A cutting projectile would be better than a low caliber handgun against polymer fabric armors, but a rifle will cut right through those anyway too. This limits your choice of targets to people who aren't geared up for a serious fight: night raids, supply stations, police, some limited ambush situations, etc.
You can get through windshields and door panels and hit a target on the other side with an arrow at reasonably close range, but not reliably. If your arrow glances off, it's not subtle. See https://www.youtube.com/watch?v=yAcWu6Y68pA for a demonstration. "Reasonably close range" isn't a good plan, though. More on that later.
Forget it. Then again, the rifleman can't shoot through these either. That's the point.
If you're fighting one person with a gun, the element of surprise is pretty much all you need. Drop your target, get into cover. If there are four or more people with guns, you're going to die. They have better maneuverability. An archer needs to make larger movements to get out from behind cover in an extended firefight than someone with a gun.
This is a little more interesting. Arrows maintain lethal force through a parabolic arc better than a bullet, which makes the effective range of an 80 lb. compound bow very comparable to a general purpose rifle like the M16. Not longer, mind you. Comparable. They both come in at about 500 yards. A heroic figure might be a better marksman at that range than red shirts with firearms, even though in the real world, the arrow is more sensitive to environmental factors like wind. Creative license!
There's almost no such thing as an extended sight line in an urban environment. You could lob arrows at groups from a safe distance, maybe try to force workers indoors, but otherwise, you're limited to the short range constraints above. Also, trying to shoot out of a window will limit your fire lines and, in turn, reduce your maximum range.
This might give you better angles for hitting light vehicles at short range. Other than that, this is just a trap. What interesting targets are going to be on foot, alone, and lightly armored?
I wouldn't pick a fight on the open plains. You'll just get shot.
There are some interesting tactical opportunities for beaches, though. You can fire from a boat into open windows or at off-duty combatants. You can catch sailors boarding, supplies leaving harbor, things like that. Basically, it provides an exception to the short range limitation of the urban environment.
The person you're describing here isn't really a soldier anymore. He can hit civilians, unaware combatants, and their support troops. He can't take and hold territory. All he really can do is cause disorder and break morale. This person is a terrorist.