i own both cameras. both are fantastic for shooting sports. if i had to pick one of them to use over the other for anything sports/wildlife related, i would almost always choose the 1d over the 5d. there are a few very important things to take into account when making that decision - frame rate, body durability, crop factor, battery life, af system. image quality is important too i suppose . . .
as for the frame rate, the 1d is 10 and the 5d is 6. that one ought to be a no brainer to anyone who has shot sports before. the faster your camera can take pictures, the better chance you have of getting the action shot you want with the facial expression you want. is 10 fps necessary? no. i started shooting sports with a 5d1 - who's frame rate is just shy of 3 fpslow.
thats the equivalent of walking 20 miles to school through the snow. i made it work and i got the shots i needed to get to complete my assignments. however, i honestly don't remember using my 5d classic for sports once i got a 1d unless it was a situation where i absolutely needed two bodies. the 5d3's max is 6, which isn't slow, but it sure isn't 10.
durability. i can tell you for a fact that the 1d4 is built like a tank. i've put mine through some pretty ridiculous conditions and it always worked flawlessly and cleaned up after assignments/vacations/music festivals/salt air. it's been rained on countless times. i did have an issue with the hot shoe not wanting to work in ettl 2 with my 580ex one time, but i thoroughly cleaned the shoe and it works again. that's it. no other issues.
since i already have a camera i know has proven itself in the elements, i haven't done anything with my 5d3 to really test it's real life durability. i can tell you for certain though, it doesn't get the same "tank" status that the 1d gets and if you held both cameras at the same time you would know exactly what i mean - one looks delicate and the other looks like it could roll down your stairs and be fine. seriously.
another big thing that sports shooters want to take into consideration is the crop factor of their camera. canon sacrificed some image quality with a 1.3 crop sensor in the 1d4 so the processor could handle 10 frames a second. if you're seriously looking for degraded image quality from the crop sensor, you'll be able to zoom in and find it - but it still looks great. but what 1.3 also means is that your 70-200 now gives you a reach of 260mm - which rocks in my opinion. however, you're 16-35 just became a 21mm and your ultra wide capabilities are gone with that particular lens - which sucks in my opinion. overall, i'd say most sports photographers would choose reach over a wider angle.
the battery life on the 1d4 is nothing short of ridiculous. i've shot over 1000 images without the battery going below 50%. at 10 fps, you won't believe how many shots you can take during a 3 hour game. you're 1d battery wont die. not even close. ill go months without having to charge that thing sometimes.
the 5d3 has a bigger sensor and in turn a bigger mirror to power and operate which means more battery gets eaten. a lot of the work i've done with my 5d3 has been on a tripod and hardly ever bursting frames, so the 5d3 battery by itself works quite a while for me, but i wouldn't go shoot something serious without an extra 5d battery. if you do choose a 5d3 as your sports camera, i recommend getting a grip - immediately. having a second shutter button is largely underrated in my opinion. plus you get room for a second battery. with a grip and two fully charged batteries, i would be surprised if most people get close to draining one of them.
there is a fantastic af system in both cameras. the 5d3 has the af system from the 1dx. i've heard lots of people complain about their experiences with their 1d4 auto focus/servo. it took me a few weeks to get my af custom functions dialed in, but after that, the servo has always been spot on. i unfortunately haven't put my 5d3 through any real servo tests since i first got it, but it was very, very impressive for a 5d in my opinion. the af menu is quite a bit different than the 1d4, but much easier to use once you get used to it.
when it comes down to overall image quality, the 5d3 wins. there generally isn't going to be any application where that image improvement over the 1d4 is going to be absolutely necessary, but it's definitely there if you look close enough.
fortunately, when i go shoot, both of them get to come with me. the 1d4 gets a 70-200 and the 5d3 gets a 16-35 or 24-70 - depending on the situation. they're a hard combination to beat.
for most people, i'd say the 1d4 is extreme overkill. do you need 10 fps? is it worth paying a lot more and sacrificing some image quality for that 10fps? do you mainly shoot fast moving subjects? im not sure what the rate for a new 1d4 is right now, but im guessing you could get a 5d3 and a nice lens for the same cost, maybe less.
honestly, i'd say a 5d3 is overkill for most photographers too. unless you need the extra fps and the 1dx af system, you could save yourself a lot of money by getting a 5d2 or 6d and invest in glass.
i know a lot of this is repetitive throughout the thread, so i apologize.