The seemingly little differences in some English words, especially when it comes to spellings, are a result of the differences in the way some English words are being written and spoken in different countries, places, etc. British and American English is the most used English in the world. And there are some differences in the way some words are spelled both in British and American English. That's why you will see some words with the same meaning but a little difference in their spellings.
The same explanation is what goes for these two words; favorite and favorite. In American English, there is the omission of the letter 'u,' and that is why, out of the two spellings, a favorite is the right spelling in American English. The same thing is applicable to British English; the second spelling (favorite) with the letter 'u' is the right spelling. Other examples are; odor and odor, color, and color.