Here are the benefits to cutting a track to a click:
1) The tempo is more even so the song can feel better as a result (more on this later).
2) Because of the even tempo, it's easier to cut between different takes to obtain a superior performance.
3) During mixing, it's easier for the engineer to time delays and reverbs to the track so they blend better.
The major downside of cutting to a click with a drummer who's not comfortable with it is that the track can sound stiff and machine like. If the song doesn't feel good, none of the above benefits matter much.
While it's easy to believe that every hit song has an even tempo, that's not the case at all. I found the following graphs on a post called "In Search of the Click Track" that had some plots showing the tempo deviations from the average tempo for a number of songs. It's pretty easy to see where machines set the tempo and where it was all human clock.
The first example is a comparison between a Police song ("So Lonely') and a Britney Spears song ("I Love Rock n' Roll"). There's a huge deviation between Stewart Copland's playing (he's always been ahead of the beat and it sure is in evidence here) and virtually none in the Britney song, which is obviously a machine.
If we look at "Ya Ya" by jazz great Art Blakey, we see that the guy is as solid as a rock and as close to perfect as a human can get without playing to a click.
Charlie Watt's playing on "Sympathy For The Devil" is all over the place, but it's always felt pretty good to me and I never would have thought you'd find a deviation like this. It's a 17 bpm difference between the beginning and ending of the song!
But Nickelback, on the other hand, is obviously playing to a click, and the time hardly budges.
The Who's "Won't Get Fooled Again" proves that Keith Moon was playing along with a sequenced synth (you can see him wearing headphones during live concert videos of the song). The thing is, he could really do it well as he still made the song feel good, even despite his unorthodox style.
So what can we learn from these charts? If the feel is good, so is the song. Tempo variations are mentally overlooked by the listener under the right circumstances, and a solid steady tempo doesn't necessarily sound boring. Each song is unique, and therefore the question of "To click or not to click" remains unanswered because there is no answer.
I believe that the difference between a good drummer and a bad one is how good his internal clock is. That is, if the drummer can play steady and even within himself (where the snare and hat always hit in the same place relative to the kick drum, despite the tempo), the drummer will be considered "solid." I think these charts prove that what makes a song is not a tempo thing, it's a feel thing. A good drummer equals a good feel, but a good drummer may not always equal a good tempo.