Is there any reason to pick bow-fishing over rod-fishing.
As far as I know it's extremely difficult to aim for fishes beneath the water surface. This makes me wonder if there is any rational reason to prefer bow-fishing.
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It seems that your question is about efficacy of the method. I'm not a bow fishermen, but I know that some plankton feeders, such as the asian carp and the silver carp will not easily take a lure or a fly. In fact, carp is one of the most challenging fish to catch on an artificial lure (although they do take bait). In that case, bow fishing seems to be the way to go. In the case of the silver carp, it jumps often and can be shot mid air. See this link for examples of how these species are hunted.
Both fish are invasive species in many waters, so bow fishing (being a lethal method) may actually be the preferred option. I'd think that people bow fishing for other species of gamefish (trout, salmon, for example) would be frowned upon in certain communities, though.
Edit: not being a bow fisherman myself, I totally missed the issue of visibility that Escoce pointed out in his answer. That means that bow fishing is favoured over rod fishing in instances where fish are plankton feeders, sabiki rigs are not legal (some jurisdiction limit the number of flies in a rig), they are visible in the water, and the target species is one that can be harvested without problems.
Bow fishing is a sportsmanship hobby like any other. If you like the idea of bow fishing, then try it.
If you like the idea of rod and reel fishing, then try it.
Which is better? Well, you can only shoot a fish you can see with the bow, and rod fishing lets you hunt for fish in places you couldn't possibly think to use a bow, such as deep water, thick weeds, water so brown you can't see this fish unless it's jumped into the air. The biggest fish get big because they know how to hide.
It's a form of hunting. Shoot a rabbit or snare it? Rod fishing you are luring a fish into a trap, bow fishing you are actively hunting.
Some say you need an entire new kit but all you really need is a specialised type of arrow. It is all down to preference really.