Both versions of the 300mm f/2.8L IS USM have extremely good image quality. Some reviewers have called each of them the "sharpest lenses Canon or anyone else ever made".
The EF 300mm f/2.8L IS USM II is almost 1 lb. lighter than the original version... just over 5 lb. vs close to 6 lb. (For comparison, your 300mm f/4L IS and 400mm f/5.6L are each around 3 lb.)
The "II" also has the latest and greatest Image Stabilization system... possibly good for another stop of assistance, also with "Mode 3, Instant IS" which only activates during the actual exposure.
Probably the most concerning difference... Canon no longer supports the original lens. Spare parts to repair it have run out. As a result, if anything breaks, it may not be possible to repair it.
Is that worth the extra $1600? It would be
IF the original lens' autofocus or other mechanisms fail, it's unrepairable and becomes little more than an expensive paperweight!
Frankly, I don't see why you need either. The EF 300mm f/4L IS USM is quite good, both alone and with the more recent 1.4X teleconverters. So is the 100-400L II! The 400mm f/5.6's image quality is superb, too... though it lacks image stabilization. Their size and weight also make them all much more "hand holdable" lenses and preferable for hiking or travel. (I've traveled with 300mm f/2.8L! It ain't easy, but it does seriously impress the TSA inspectors.
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Much as I like "big glass", I now only get out my 300mm f/2.8L when absolutely needed for challenging, lower light conditions. As DSLRs have offered higher and higher usable ISOs, that's been less and less often. I found I was stopping the 300mm f/2.8 down to f/4 or smaller often, anyway... looking for sufficient depth of field. I now use my 300mm f/4L or 100-400 II
FAR more often. I use those lenses hand held much of the time. With the 300/2.8, which I don't use enough to justify upgrading to the II, I nearly always use a tripod with gimbal mount. It's hand-holdable briefly... But when I'm photographing a sporting event it's often an all-day shoot. It's often similar with wildlife, too. I sometimes put the 100-400mm II on a tripod w/gimbal (it's, what, 4.5 lb?).
I cannot recall the last time I turned off IS on any of these lenses. In fact, on many of my lenses with IS I have the switch covered with gaffer tape to be certain I don't accidentally turn it off! Even when it's not needed because I'm using a fast shutter speed, I find that it can stabilize the image in the viewfinder to some extent, which I find useful when trying to track moving subjects.
The 300/2.8s and 100-400 II all have "self-cancelling" IS that detects lack of movement, such as when locked down on a tripod, and turn IS off automatically themselves.
The 300mm f/4L IS doesn't have this type of IS (neither did the original "push/pull" 100-400mm and three other specific, but now discontinued Canon IS lenses)... That form of IS can "freak out" on a tripod, actually causing "shake blur" when there's no movement to correct. HOWEVER, these lenses are ones I use hand-held 90 or 95% of the time anyway. On a monopod, IS can stay on. There's plenty of movement to correct. Even on a tripod with a gimbal, there's no need to turn it off. It's only when a lens is fully locked down in some way, that five of the Canon lenses (the 300mm f/4L IS the only one that's still in production) might require turning IS off manually at the switch.
All the Canon IS lens manuals suggest turning it off on when using them on a tripod and/or at higher shutter speeds. Howerver, according to Chuck Westfall, Canon USA's former tech guru, this is merely suggested to save a little battery power. (Very little... IS doesn't seem to draw very much power. I notice virtually no difference in the number of shots I get using IS and non-IS lenses alongside each other on identical cameras.)
Nikon users feel that their VR system tends to slow autofocus slightly, and there seems to be some evidence to support that belief. Personally, after using a variety of different Canon IS lenses for over fifteen years, I feel it's the opposite... that Canon IS actually helps AF perform better. Keep in mind that each manufacturer's stabilization system is a unique, patented design. So even though they are all intended to serve the same purpose, it's likely there are some differences between them. Also, Canon had a pretty big head start, putting stabilization in lenses 8 or 10 years before Nikon started doing it too.
You may want to turn off IS when shooting video or possibly if trying to be very precise with the framing of your image. That's because IS can cause a little "jumpiness" or "slow drift" that might effect videos or subject framing.
Both versions of the 300mm f/2.8L IS USM have extr... (