I thought there was almost no difference, but to be exact, there is.
 
In kanji, it may be easier to understand the gap.

bushi:武士
武 (bu) means “a fighter with a weapon”
 
士 (shi) means “a man”
 
So in a strict sence, bushi means “a fighting man with a weapon ”
 
samurai:侍
侍 means “one who serves”. Samurai served to a noble or a imperial family.
 
　People (including Japanese) tend to call a man with katana swords “samurai”.

 
But it would be more accurate if we call him “bushi”. 　
 
Even japanese people are not aware of these difinitions, so nobody points out which word you should use.