Generally, cheap trailers break whether they are utility or enclosed.
Most Cargo / Utility trailers aren't designed for off road use.
They are designed for paved road use.
Suspension, Tires, and Protection are needed for most trailers for hunting application, especially after a few years of use.
With a lower entry cost, the utility trailer is more economical to modify for hunting applications.
It might be worth it to just park down lower on the mountain and use the side by side to access the areas rather than beating on your trailer, which most are designed to do.
Double Axle > Single Axle in both.
Utility trailers generally tow easier than cargo trailers, especially in wind.
Weight distribution is important in both styles of trailer.
Cargo trailers are more modifiable than Utility trailers, you can turn them into campers and/or have wall storage for all your things.
With Cargo trailers you can't see if things have shifted during your travel and you can open the door to a surprise.
Depending on the height or width of your side by side, you may need a size wider or taller in Cargo trailers.
It can get tight fast if your cargo trailer is too small.
Cargo trailers will require more maintenance over time. A leaking roof will be a bad time if discovered on a camping trip.
A few things that will help in either.
Shocks - You will wish that you did this sooner for ride quality.
Stabilizer Jacks - Allows you to disconnect the trailer without having all the weight on the front jack
E track - Tie Downs, Storage, etc. A very versatile system that is easy to install.
Tire Changing Ramp - The easy way to change tires