Loss of Colour Detail After Premiere Pro Render
First, Youtube re-encodes anything you upload to it's own streaming format(s), creating multiple down-res'd versions from the file you upload. 

So it's possible that:

1.) OBS is outputting a higher-bit rate or better codec than what you're outputting using Premiere. 

2.) Browsers ignore for the most part (Chome), color profiles for video and photos, this can cause a lot of frustration wondering why colors look different in Photoshop/Premiere vs in Chrome. That said:

3.) From Bridge make sure your Color Settings are synced across all programs under the Edit Menu. 

4.) Set to Web Color sRGB. 

In the end, the hard truth is that you CANNOT control the "end result" in terms of how your video will look on youtube on any given computer. Because people have different monitors, different browsers, their monitors may be setup to non-sRGB profiles, etc. 

It's very much the same as handing someone a Blu-Ray, that you've quality checked on your studio reference monitor to get the tonality you're aiming for. You have no idea what TV they will use when they take it home, and what Picture Profile they will have set, which can drastically alter the image they see. 

Obviously however, whatever OBS is outputting is being converted into a format that Youtube is re-encoding to a file which you are more pleased with. The only solution is to try different output settings from Premiere. But honestly, it's a bit of a fruitless endeavor, as you have no control over how other people will see it; due to browser, color profile, monitor setup, etc.