Groove or Chops?

Like most things in drumming, this is an issue swamped in debate. Let's have a look at both sides:

Groove
Groove is based on the feeling of the music. Regardless of technical ability, a drummer who plays a groove fits perfectly with the music and provides a solid beat. Listen to any song, and work out the beat, regardless of time signature. This tempo, like the click of a metronome, is the basis for your groove. Funk music relies heavily on groove, and drummers like Chad Smith (I know I feature him a lot, but there's a good reason why), who plays a mixture of rock and funk, is someone I respect for his brilliant ear for the groove. Have a listen to him playing the most basic of rock beats and see how he fits with the bass:




Chops
Put simply, this is just technical ability. There are some incredible drummers out there who can play amazing things. This sort of drumming is incredible to listen to with or without a backing track. Just look at Martin Škaroupka here, for instance:



Both
You might have seen this coming, as both of the above drummers have incredible chops and a feel for the groove (in different genres - let's not get into a 'who's better?' argument), but I'm saying that a mixture of both groove and chops is important to be a versatile, exciting drummer. One drummer who is world famous for his amazing ability is Tony Royster, Jr. Watch this if you don't believe me:



But, amazing skill put aside, he also plays for Jay-Z, and plays some solid, simple beats in this gig:



Here we can see that no matter how amazing and talented you are, there are some situations where you should only play a basic groove. Don't try and over-play to prove you're a good drummer. If you're good, people will notice; there's no need to play your most complicated fills and patterns when all the song needs is a sturdy 4/4 beat.