how much moisture protection do you want, that's the big difference, I'm going to paint, seal then paint but the shed I'm painting is a closed space and the humidity gets high at times, then the moisture gets into the particle board and takes forever the dry out, I'm getting a better vent for air flow this spring too. stain just stains
I used a waterproofing stain on the exterior of my coop. It was by Behr (Home Depot) and only lasted 1 winter. In the spring it was flaking off. Granted, we had a tough winter that year, but come on. Not happy. I ended up recoating it with linseed oil/mineral spirits (I like the "wood" look vs. paint) and that has worked well. It has to be redone annually, but only takes an hour or so. I painted the interior with exterior semigloss. Walls and floor. Great for cleaning. If you decide to go with paint, make sure you prime first.
If you're talking about the exterior, probably either would work. They make some really pretty colored stains these days.
After using stain on my original coop (circa 1993) I would never use paint again. It has only been redone once (2001) and needs to be done again, however all you have to do is clean & restain, there is no peeling paint. The trim on the original coop was painted in white latex and that peels off and has to be redone much more often. The stains come in many colors, the protection is great. My coops are T-111 board if that makes a difference.
Paint will protect the wood better and for longer; however it is also more of a nuisance to redo when the time comes, as you really need to get off ALL the stuff that is even slightly at all loose, as opposed to a quick sand and re-stain. Personally I am still a "prime and paint" person for most applications, but a waterproofing stain (tinted sealer) is not a totally unreasonable option if you are a different personality type than me
One disadvantage of stain (tinted sealer) is that it highlights all the imperfections in your wood and differences in color/type between the differetn pieces you used. Even a very dark stain will reveal these things somewhat. Whereas a paint makes it look all of a piece. If you are aesthetically picky, therefore, paint is sometimes the best option for that reason too.
Well, I would go for paint.
You get paints that are waterproofers/sealers/gloss paint
How is your budget?
If you have a tight budget then go for the cheaper one.
If budget is not a problem choose either one. The number one thing is, is it good quality paint/stainer that you choose?