I often get the questions about which frame rate to shoot and there seems to be alot of confusion out there about the difference between 24fps and 23.976fps. There is a ton of technical stuff on the web about this and if reading some of it doesn't make your eyes cross, then you might find what you're looking for. Wikipedia has numerous explanations, and while the Wiki articles are great for all the heavy technical mumbo-jumbo, I don't feel that it really answers the main question, which is about the ascetics differences between the two frame rates, which in actuality isn't much. For the record, I'm referring to progressive images here, not interlace (who shoots interlace any more???).
I've discussed this with numerous post houses over the past few years, and with their help have come to somewhat of a conclusion - IT DOSEN'T MATTER - but, keep it consistent, and everyone will be happier. Pick a frame rate for your project and stick with it as much as you can, ascetically there is no difference between 23.976 and 24 (see frame rate math below). Remember, I'm not talking about effects type shots where you over or under crank the camera to achieve time compression or expansion, that's a different frame rate consideration.
HOWEVER, if you are incorporating 24fps FILM in your project then you should scan/transfer at 24fps and let that be your standard frame rate for the project. The same holds true if you are going to film the project out, use 24fps. The main thing is to keep everything consistent, so there won't have to be any pull up/down conversions (if you're having trouble sleeping, reading up on pull up/down is a good cure), which could lead to artifacts, and slow down your workflow. Otherwise there is no really compelling reason to shoot at 24fps vs 23.976fps. By the way the DCP standard allows for 23.976fps so that should not come into play.
If your show is going to be broadcast, you know sent out over the airwaves by a transmitter and received by an antenna, then the television station will want the project in 23.976 because it just works out better for timing.
Now edit, and export using the same frame rate you shot with, and things will go much smoother. That's the perfect world version, of course we don't live in a perfect world. By the way whatever frame rate you choose to shoot with can always be converted to something else, within reason, but image quality may suffer.
Geek Stuff - The Math of Frame Rates:
The .004fps difference between 24fps and 23.976fps equates to a very small fraction of a second increase of shutter speed, and accordingly, motion blur, which is why we shoot at 24fps as opposed to the NTSC standard video frame rate of 30fps (29.97). 24fps = 24 frames being shot in a second, that's .04166667 second per frame, assuming a 180 degree shutter which exposes the media to light for half the frame rate time, the exposure for 24 fps is .02083333 second, or 1/48 second. Now, 23.976fps = .04172404 second per frame, and at 180 degree shutter - .02086202 second exposure. Basically you have to get to the 5th decimal place to find a difference in the exposure times between the two, and rounding to the 4th decimal place makes a 1/10,000th of a second (that's 1/10 of a millisecond) difference... Aesthetically they will look the same.