As stated before...IS only counteracts the movement of your hand. Meaning, in a low-light situation you have to have high shutter speeds to "freeze" the subject in any way if that subject is in motion. If you are shooting a stationary object in low light, that is when IS becomes useful.
The big question is, what do you want the lens for? The 70-200 F4L is one Canon's best and is certainly their best value "L" lens along with the 17-40 F4L. But in order to get high shutter speeds at F4, you need really good light. Even outdoors with overcast skies will start putting your ISO 100 shutter speeds lower than 1/200 second, and if you are shooting at 200mm, you want to be faster than 1/200 second. So it's a great lens for the outdoors on sunny days, otherwise you will be shooting a lot north of ISO 400 and plan on leaving it on ISO 1600 if you take it indoors.
The 70-200 is very sharp and produces very good colors and contrast. It has that red ring. It zooms and focuses internally and is weather-sealed. It also has super-fast ring USM with full-time manual focusing. The 70-200 is also slightly cheaper than the 70-300. But it's also white and therefore attracts a lot of attention.
The 70-300 has an extra 100mm of reach. It is black and is more discreet. It of course offers IS. But it costs slightly more, it's not an "L", it telescopes, and it only has micro USM. The front lens element therefore rotates during focusing and makes it difficult to use circular polarizers. The 70-300 also lack F4 throughout the range. At 300mm, the best you will be able to do is F5.6. In terms of optical quality, I would say the 70-300 offers 90-95% of what the 70-200 can do, all things being equal.
So it comes down to what you need the lens for. If you need something for sports or low-light photography, neither is really ideal. If that is what you need, then take a look at the EF 135 F2.0L and EF 200 F2.8L prime lenses. If you just want a good telephoto for sunny, outdoor work, then lean towards the 70-200, especially if you carry a tripod or monopod. If you need a versatile lens that is decent for both indoor and outdoor work and you know you won't be carrying a tripod or monopod, then the 70-300 is probably the better bet.
At the end of the day, the "EF 70-200 F4L versus EF 70-300 F4-5.6 IS" debate is one of the most fiercely contested debates in photography circles. Two lenses, similar range, similar price, but does one choose L or IS? People do rip their hair out!
What did I do? I bit the bullet and spend $1000 on the 70-200 F4L IS and couldn't be happier. Actually, a 70-400 F4L with the same price, size, and weight would be more ideal but oh well.