Is the Canon 1D MK III a +1 in everything from the 1D MKII N? (for sports)
The 1DmkIIn is a great sports body. It doesn't fail to get the job done. But like all technology that has been superceded, it can't keep up with the newer technology when that newer technology has been designed to surpass it (unless you never utilize it in a situation where the newer technology has the advantage).
To answer your question regarding the mkIII being +1 in everything from the mkIIn; That would be a solid YES!.
Keeping the same crop factor (APS-H sensor) you jump from 8 to 10 megapixels giving you a bit more cropping range. Not essential, but usually beneficial.
Battery technology changed from NiMH to Li-ion. I've been able to take over 11k images on a single charge. Average on the mkIV is 6000. Average on the mkIII over time has been less than 11k per charge.
Menus and Menu selections: The mkIII has many more options in its menus and does not require you to press two buttons at a time to make changes to major settings. The 1D, 1Ds, 1DII, 1DIIn and 1DsII share this two button to change settings requirement.
I still use my mkIII right alongside my mkIV's and reserve the 1DmkIIN for outdoor daytime events. I would shoot with it at night under the lights if I had to, but am thankful I don't have to.
Your highest ISO on the MK III is 6400 and only available if you've set C.fn I-3 to allow it.
If you pick up a 1DmkIII you'll definitely enjoy it.