Is battlefield three better than call of duty three

This comes down to a matter of opinion, but here is my unbiased view of the two games.

Singleplayer
Battlefield 3 has a story that you can easily follow as it's not too dependent on other games in the series.

Modern Warfare 3 has the plus of continuing the story from the previous 2 games. If you have never played the other games you will not be lost, however.

Both campaigns are fairly short, but I would put Battlefield 3 as being a little longer.

Multiplayer
Battlefield 3 has larger environments and, debatably, a more strategic online experience.

Call of Duty: Modern Warfare 3, assuming that is what you meant, is more of a run and gun game with smaller, more urban maps.

Both feature leveling systems which let you access more guns and equipment as you level up.