You can get some idea comparing online test photos:
Tamron 100-400mm at 300mm versus your 300mm PF:https://www.the-digital-picture.com/Reviews/ISO-12233-Sample-Crops.aspx?Lens=1178&Camera=453&Sample=0&FLI=3&API=0&LensComp=1040&CameraComp=614&SampleComp=0&FLIComp=0&APIComp=0
Tamron 100-400mm at 400mm versus your 300mm PF with 1.4X TC:https://www.the-digital-picture.com/Reviews/ISO-12233-Sample-Crops.aspx?Lens=1178&Camera=453&FLI=0&API=0&LensComp=1040&CameraComp=614&SampleComp=0&FLIComp=1&APIComp=1
Tamron 100-400mm versus Nikkor 200-500mm:https://www.the-digital-picture.com/Reviews/ISO-12233-Sample-Crops.aspx?Lens=1178&Camera=979&Sample=0&FLI=2&API=0&LensComp=1035&CameraComp=1210&SampleComp=0&FLIComp=0&APIComp=0
Unfortunately, the above comparisons aren't perfect... The Tamron lens test shots were done on Canon cameras, while the Nikkors shots were done with Nikon cameras. I've selected the most similar resolution full frame for the best possible comparisons: 1DsIII (21MP) and D3X (24MP) in the first two cases and 5Ds-R(50MP) and 850D (45MP) in the third.
I was surprised how well the Tamron seems to hold up compared to the 300mm lens. When the 300mm is being used without a teleconverter, it's a full stop faster than the Tamron.
The Tamron also seems to hold it's own against the Nikkor 200-500mm at focal lengths they share (both suffer a little when shown on very high resolution cameras). With both at 200mm it's the same f/5.6 aperture as the Nikkor... at 300mm and beyond the Tamron is 1/3 stop slower (f/6.3) than the Nikkor (f/5.6).
The Tamron's weakest focal lengths are toward its 100mm end.
The above website also has means of comparing distortion, vignetting, flare, etc. It's pretty thorough testing, though there are limited review of brands other than Canon, its primary emphasis. There's a good write up on the Tamron though it is the version for Canon. There isn't a full review on that site for either the 300mm PF or the Nikkor 200-500.
There is option of fitting a tripod mounting collar (sold separately) on the Tamron lens. Personally, I'd spend the extra for that because it would be nice to have at times, I'm sure (I often use the collar on my Canon 100-400mm, putting the lens on a gimbal and tripod). Also, in recent lens models, including the 100-400, Tamron has been been incorporationg an Arca-compatible dovetail into their tripod collars.... That's a thoughtful feature I wish more manufacturers would do, so a separate lens plate wouldn't be needed.
The Nikkor 200-500mm's tripod mounting foot isn't very compatible with typical lens plates (AFAIK, only Hejnar makes a special one for it). So when folks want to use it on a gimbal or want Arca quick release compatibility, many opt either to replace the foot or even the lens' entire tripod collar. Kirk Photo, RRS, Hejnar Photo and others offer various options, adding cost of course.
Compare size, weight and cost: https://www.bhphotovideo.com/c/compare/Tamron_100-400mm_f_4.5-6.3_Di_VC_USD_Lens_for_Nikon_F_vs_Nikon_AF-S_NIKKOR_300mm_f_4E_PF_ED_VR_Lens_vs_Nikon_AF-S_Teleconverter_TC-14E_III_vs_Nikon_AF-S_NIKKOR_200-500mm_f_5.6E_ED_VR_Lens/BHitems/1362803-REG_1111442-REG_1052071-REG_1175034-REG