Same:
Both 忘 and 忘记 can mean forget something(somebody) or forget to do something.
Different:
1. 忘记 stresses that something slip from your memory.
2. 忘了 can put at the end of a sentence but 忘记 could not. There must be something follow after 忘记.
ie: Don't forget it! 别忘了!
I'm sorry, I forgot. 对不起,我忘了。
3. The usages of the two words are different.
Forget to do something.
ie: I forgot to do my homework.
我忘记写作业了。
我忘了写作业。
ie: I forgot my passport.
我忘了带护照。
我忘记带护照了。
Forget something or someone.
ie: I won't forget her.
我不会忘记她。
我不会忘了她。
ie: I didn't forget you.
我没有忘记你们。
我没有忘了你们。
Ask someone don't forget something. (usually don't use 忘记)
ie: Don't forget your bag.
别忘了你的书包。