They both basically do the same thing only crosfire is for AMD cards, sli is for nvidia cards.
Remember that its always generally more beneficial to buy a better single card than to use these.
Advantages of AMD cards, Cheaper, Traditionally Lower power, more of a no frills simple approach to things.
Advantages of Nvidia cards, Typically better performance though you can pay a lot more for this, more features like g-sync/sheild. I find the drivers a lot more reliable though game performance is more driver/optimization dependent.
I've always had bad experiences in trying to get AMD drivers to install and am currently in love with my 2 GTX680 cards, so if buying today I'd go with a 780ti anytime
http://lifehacker.com/the-difference-between-nvidia-sli-and-amd-crossfire-1555235577