Personally I prefer the cross bow but there's no real advantage to either when you consider the pros and cons and just have to pick which works more.

Bows shoot faster than cross bows and are quieter. That's pretty much it from what I've seen. I guess since they shoot at an arc a little they have more range but frankly I don't see a difference and I shoot just fine even a mile away with my crossbow.

Crossbows do a little more damage but mainly their advantage is a much harder hit where with Power Shot they stagger everything almost all the time. They are also an ammo weapon meaning they are actually faster than a bow when it comes to the first shot at least which is usually most important. Also cross bows have a built in ignore armor effect too. People have said this doesn't make a difference but I hugely disagree. It doesn't matter when fighting creatures, dragons, mages, etc but u always notice a substantial increase in damage against any armored humanoid opponents. It DOES make a difference against the right enemies.

As for bows being quieter I find that makes no difference at all once you get muffle and your sneak skill is higher.

Now for me I prefer cross bow just because it prefer the look and feel of an ammo weapon. I remember when I first played Dawnguard and Isran handed me the cross bow. Initially I was like what but when I armed it and shot I thought "Whoa! It's like I have a gun now!"

But in the end I don't think there's a huge difference in over all fighting power compared to a crossbow and bow. In the end it's more likely just about which one looks cooler to you. And for me it's hand down the crossbow.

While it's good for that first shot, that couple extra seconds it takes to reload a crossbow can be crucial, especially when you have something like a Legendary Dragon bearing down on you. The advantages really just depend on the situation you're in, I'd say. I've been in scenarios where the crossbow was much preferable, but I can say that the reload time has gotten me into some near-death experiences, in which case I prefer a regular bow.

True but you can have the same situation compared with a bow such as when that first shot matters or when an enemy is upon you where a few more staggers might have helped. And frankly I don't have any issues with dragons since I stagger them most of the time before they have a chance to do any breath attacks.

Like I said they both are overall equal to me

Of course I think a bow is still better for sneaking. Even though the noise doesn't affect sneaking in my view the quicker draw means the possibility of more sneak attacks before detection and in that sense it does way more damage.

But my character is not a sneaker so that doesn't mother me. His main fighting skills are two handed, archery, and conjuration. I always prefer the straight up warrior but always with some magic (two dremora lords just makes ANYONE that much stronger in combat). I'm not much for sneaking. I just enjoy charging in and so occasionally I charge in with a crossbow and make quick work of everyone while my dremora lords clean up the mess.

So I guess I can understand why that extra power hit matters for my playstyle. Crossbow seems better over all for the non sneaking warrior who still wants ranged ability (and doesn't use destruction magic).