Downsides?

Header wraps hold moisture against the header and can lead to pre-mature failure due to rust. Other downsides are that you have to find a way to secure the ends of the wrapping, usually safety wire is the common choice since it can be done tightly if you have the correct tools for it. Otherwise some elbow might and other wire can take care of it. It's also a time-consuming process usually, and the header wrap can get pricey by the time you buy the amount that is required for the job.

Upsides?

You keep heat energy in the exhaust, which helps exit velocity. Since more heat stays in the exhaust, less is transmitted to the engine bay.