I don't understand the use of a definite article in this phrase. Isn't the meaning the same in both cases?
the benefit of the doubt
This expression is used when there is an existing doubt that is being referred to. The doubt may be explicit or it may be implicit but all parties concerned will know that there is some specific doubt. The use of 'the' indicates that specific doubt.
The benefit is usually belief or trust. In a competition it could additionally be the awarding of points.
A: "Did you steal the money?"
B: "No I didn't"
A: "Well all right, I can't prove it so I'll give you the benefit of the doubt."
In the above highly simplified exchange, A had a suspicion (a doubt) about whether B had committed an offence. Because A cannot prove anything, the benefit is that A will not prosecute (or jail or punish) B.
the benefit of doubt
Here we seem to be saying that there is some benefit in doubting. That is a strange concept. It is more difficult to come up with a plausible explanation as to what the benefit might be. Here's the best I can come up with.
"Why do you always avoid watching the news when the football is on later?"
"Because I don't want to find out the score before I watch the match."
"So the benefit of not knowing is that you can enjoy the match more?"
"Yes, the benefit of doubt in such circumstances is that I can watch as though the match is playing live."
That is a plausible scenario and lots of people do that. However, the standard expression 'benefit of the doubt' would be inappropriate (or at least very confusing) in that case.
As written, "the benefit of the doubt" employs parallel structure, which makes a phrase more musical and more memorable (as does alliteration). Also, by making "the doubt" a specific one by using the definite article, the speaker is referring to the doubt that the subject is guilty, not to all or any doubt.