Okay, besides the emotional preferences, like "in close touch with nature", fun or personal preferences (and so on), is there any rational reason to pick a bow over rifle?
I mean things like "it's quieter and that's good for xy" for example.
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Other than the stated back to nature (if you can count it as that as most hunters use compounds) it is generally considered both quiter and cheaper.
It is a lot cheaper to buy a bow and arrows than a silenced rifle. Especially as, if you are lucky, you can reuse arrows.
It is down to preference however, as to what a person wishes to use.
I find bows are more challenging. Plus, in Ohio for whitetail deer, the archery season last from end the of September through February. The gun season lasts only a week and then an extra weekend. So comparatively, there is a lot more time you can spend in the woods if you're an archer.
For the pricing bit of archery vs guns. Well, my bow (Matthews Z2) cost me 2x what my shotgun (Remington 870 express) cost. Of course, my buddy just bought a turkey shotgun that dwarfs my box cost so that part is all a matter of preference. Arrows themselves are probably about 2-3x the cost of each sabot shotgun slug. Then you need to tack on another $10-20 for broadheads for your arrows. Now yes, arrows can be reused where slugs cannot. Accessories are probably similarly priced for each item when all accounted.
I personally think it is all a matter of preference and how long your seasons last. Plus, it is fair easier (and cheaper) to do more target shooting with a bow than it is with a gun. Even if you're reloading rifle ammo, you're still gonna surpass the cost of arrows rather quickly. People may be recoil sensitive so they may go archery instead. Others may be felons and not able to buy a gun. We have background checks for guns, but you can pick a bow off the shelve here. As for the noise part of it, should you really be looking to fire multiple shots? Guns make it easier for multiple since they're quicker (mostly), but I don't think you should go into it with the notion that you'll want to fire multiple shots.
You didn't mention your intended use - so lets go with it;
Hunting; - longer season - potential access to different game - when you shoot it, you don't scare off every animal within ear shot - if you manage to kill something, you are one bad mother
Targeting; - cheaper to re-use arrows - you can shoot most anywhere (park, back yard, etc) - if you manage to hit something, you are one bad mother
Survival; - you can repair a bow with elements from the wild - you can make your own arrows - living off only a bow makes you one bad mother
General; - uncle Sam seems to think its OK to run around with a deadly bow in your hand vs. a not so deadly 22. - owning a bow makes you one bad mother.
You get to go in there BEFORE the Rifle season begins so they are less pressured!
Your shot opportunities at your spots are close distance anyway.
With bow you will usually be able to take at least two shots before they bolt, since it's so much quieter.
Getting a bow kill (from the ground) gives you way more bragging rights.
NOTE: when you miss, it often can happen that you'll lose the arrows as they dive under the soft earth and oak leaves. A good carbon fiber arrow and broadhead is EXPENSIVE! For me they are like around $17 bucks a pop! So be sure you take that into consideration in your planning.