I've rented both. The 400 is a bit sharper, but the 300 combined with a 1.4x TC would be ok...however, best of all would be a 100-400 zoom. It gives you the convenience of zoom, it gives you the same amount of IS as the 300 (2 stops)...and it very likely is close to, if not equal to, the sharpness of the 300 with 1.4x converter. The 300 alone is a bit sharper and a lot faster aperture than the zoom.
It really depends also on what you will be shooting, in how much light...and how fast the subject is moving.
The only downside to the 100-400 that I have seen, is that its zoom "pumping" action tends to pump dust inside the lens. If you will be shooting in very dusty conditions, that alone would make me not want that zoom.
The 400 Canon prime lens, is difficult to use in less than very bright light, due to the lack of IS. I was however able to make it autofocus accurately in extremely dark light, but my camera's ISO quality wasn't up to the task.
I decided to order a Sigma 120-400 just today. I think, assuming I can get a good copy (and based on the pictures I have seen online), that the sharpness will be very comparable to the Canon zoom, for half the price. If however, I can't get a copy that doesn't have focus issues or other problems, I will work toward getting something else.
I hope Sigma or Canon will make a new lens category at some point. Such as an f/3.5 300, an f/5 400, or a zoom with limited range but fast aperture, such as a 250-450 f/5, or 200-350 f/4.5...all of which with stabilization. It's kind of annoying that telephoto lenses go from the $1500 range, to the $3500-$13,000 range...with nothing in between.