Until one year ago, for more than 15 years, this was my wildlife lens, used more often than not paired with TC-14E one stop down from wide open. I have never used the Sigma lens so I cannot comment on that. I suspect the example below is the above combination based on the aperture used (Exif does not compensate the for the TC). Keeping this lens steady when hand holding is more of a limitation to sharpness than resolving power, I often used it with a shoulder/hip tripod. There was a noticeable difference to the 300mm PF at 100% view when I tested it at ca 4-5m range on a 24Mpix sensor, however when viewing images at normal size it is hardly noticeable. The major difference is that it may sometimes show troublesome purple fringing in the blown highlights (for instance the sun reflected in the eye of an animal) that must be corrected in post. The 300PF is stellar in this respect (and sorry for the pun, also so in night sky star captures).
Slight crop of D200 capture with AF 300mm f/4, one stop down, likely with my modified TC-14E, ISO 400.
I wrote here that I will never sell my AF 300/4 due to the robust build, but I must admit I have had second thoughts about that after getting used to the light weight and size of the 300mm PF...
Edit: I hardly ever used this lens with AF, and with the TC it was not an option.