In some cases 'defeat' and 'beat' mean basically the same thing - to be victorious over somebody or something. 'Defeat' is more formal, however.
Also, if we are talking about a battle or warfare, you would only use defeat.
"The Norman forces defeated those of Harald at Hastings."
Sports:
"Detroit beat Houston yesterday, 4 to 1" = "Detroit defeated Houston..."
In the case where there is some race to a goal, you would only use beat.
"Thomas beat John to the finish line."