I agree with a lot of what is said here, but as to "what is the more professional sound?", there's really no right way.  Just to add my own 2cents to the mix (no pun intended), is to record for the song, not for the kit.  If I'm tracking a jazz song, I'm probably going to be wide open, if I'm doing pop or rock, I'll do more gating and isolation.  But I will say that to me a drum kit is not a collection of individual instruments, I view it as one collective instrument.  Too much of one thing is usually not a good.  A super blended kit could sound muddy and without character and attack, where a super isolated kit could sound disjointed, unnatural and worse than MIDI drums.
 
Recording acoustic drums is a Pandora's box of choices, what type of kit are you using (maple, birch, other)? What head choice? Why cymbal choices? What mics are you using? What mic placement are you using, etc., etc., etc.  However nothing to me sound better on a session than a live acoustic set, with a human being playing it.  Then again one of the worst sounding things is a live acoustic set, with a human being playing....badly. :)
 
One more tip, that I got from watching a Thomas Lang video in his home studio.  He always records with something like 24 mics all the time, but he does not use all of those mics in the final mix.  IOW he captures everything and has every thing already setup and placed, and ready to go, but when he mixes the song, he choses the tracks that best fit that song, be it 4, 12 or all of them.  I currently only record on 8 channels but I'm going to be moving to this method, because rather than after listening to play back and deciding that my kick mic needs to be closer to the batter head, using 3 or 4 mics on the kick in different places, I can just select which one, or mix of ones that sound best and not have to re-do anything.
 
Experiment blended and isolated, and somewhere inbetween, and save everything you do to a track template so it can be easily recalled.
 
Cheers!

Personally I feel that without a little bleed, the individual drums can sound a little disconnected from each other.

Etmos
Eric covered most of what I probably would have said... Also sort of depends on the style of music you're recording... Really fast precision prog-rock / metal would probably require more gating / cleaning, than say... a Tom Petty record or something... But half the fun of recording drums (if you actually consider that to be FUN), IS the experimentation really.  Just mess around with stuff, and see what works, but like Eric said, the more mics you have, MAKE SURE YOU DON'T IGNORE PHASE ISSUES. 
 
But I'm curious, just because I'm a drum recording / microphone geek, what exactly are the mics you're using for everything?  I don't generally get a chance to talk microphones with people on "digital recording" forums these days, so my ears always perk up when I hear someone say they're actually miking things these days.  (I mic EVERYTHING I possibly can...)  SO YEAH!!!  MICROPHONES!  WOOOOOOO!!!  *high five*  Sometimes I even mic the studio monitors, just to experiment with random tones & colors. 
 
Microphones...
 I am using a mixture, Audix D2 high tom, D4 Low Tom, D6 kick, Snare top Audix I5, snare bottom sm57, one overhead Rode NT1, Room mic XL990.  I have 2 NT5 on order for overheads
*sigh*
 
So dreamy...

The style of music I am recording is smooth Jazz R&B
 

BMOG
The style of music I am recording is smooth Jazz R&B
 

Yeah, for "smooth Jazz" I would probably use less mics, and allow more bleed through the mics you DO end up using.  For R&B, I guess that (to me) is sort of a wide genre in itself.  If you're doing stuff more along the lines of Usher, or "Dance / R&B / Pop" type stuff, then I would probably use a few additional mics, minimize bleed, and then maybe go nuts with effects (just to experiment with different sounds and stuff).  But if by R&B you mean "R&B more along the lines of a smooth jazz recording" then I would stick with the less mics, and more bleed. 
 
But really, experimentation is where it's at with drum recordings.  I just admire anyone that chooses to mic stuff instead of instantly going to samples these days.  I started with drum machines & samples for everything, because I couldn't afford decent drum mics (or a drum kit, etc...)  But now that I have a full kit, and crazy mics to choose from, and all of that stuff, I would have a hard time going back to using samples, they just sound unnatural to me (even "professional recorded ones" like BFD & Slate, etc...)  I still use those sometimes to experiment with layers, but for the most part, miking a live kit, and practicing until it's perfect (in your idea of perfection), that's where it's at man!

I'm a bit of a Steve Albini head...
 
This is an interesting video on Drum Recording from the man himself.
 
http://www.youtube.com/watch?v=FvLuP4Kya8U

There's a great article in this month's SOS about it. Or rather, an in-depth answer to someone asking the same question.

I'm no pro, but I record my drums at home for all of my compositions into Sonar. I spot mic all my drums (plus snare bottom), have a pair of overheads, individual hihat mics (I have 2 pair of main hats) and sometimes use a room mic. The kit is big (you can see in the picture to the left). I don't really subscribe to the isolation mantra. I'd like to think I get pretty OK results, but I'm really just doing it for my own enjoyment and learning. I'm finding myself adjusting my playing more than anything, attempting to balance things that way at first (lay off crashes, not so hard on hats, a bit of extra meat on the toms). Plus, I've really seen that tuning has a huge impact. The first couple recordings I did, I used a bit of moongel on the toms to try to get a particular sound. I ended up having to gate the crap out of the toms because when I brought up their levels to match the rest of the kit, the sustain was so long that it was unbearable. Since then, I've recorded the toms wide open (short of a single moongel on the floor toms) tuned a bit higher and it seems a lot easier to mix. A couple tracks I've done a few esoteric things as well (I went for a very specific sound in one track - taped washcloths over all the drums and played some Hal Blaine style tom melodies and had a really cool Don Henley "papery" sounding snare drum that I think worked well. Another track I tuned all the drums (including the snare) really low. I'm still working on tweaking that one, though. I may go back and rerecord it because I'm having a hard time getting what I have in my head down on tape (figuratively).
 
I'm still learning about this stuff (been really only doing it about a year now), but I'm really happy with the results I've been getting. I do have a pro engineer friend that I've bounced a few ideas off of, but even he has said that my mixes are getting pretty good. I do a variety of styles, whatever I'm into that particular week.

Hi Jeebus. Finally a familiar face!

hey dude - I thought that was you.