IMO, usually not. The key is the digital/analog converter and its quality, or lack thereof. Until it hits that, it's all just 1s and 0s from the CPU, and it doesn't matter what kind of sound chip you have.
Now, mobo sound isn't known for awesome D/A converters. But if you're working with headphones, even fairly good ones, you won't notice most of the suck. It's certainly no worse than the sound you'd get out of an iPod or most consumer-grade CD players. Given the spaghetti mess behind most computers, there's a pretty good chance your headphone is running up, down, in between and wrapping around a bunch of AC power cables and other interference sources, which is probably hurting you way more than low quality onboard sound could be.
If you have external speakers (surround is pretty popular for gaming/HTPC applications) then you should just be doing a digital out to the receiver/amp. It'll be a better quality D/A converter than what's on your motherboard. Circumvents the problem neatly, and you can plug your headphones into those.
Standalone D/A converters are available, (essentially a USB soundcard) but tend to be overpriced "audiophile" crap. (No better than what's onboard, imo., but with gold connectors and fancy names for stuff.) But if you've got a cheap laptop that's inadequately shielded internally (an HDD or system fan causing an audible click/whine/hum on the audio port isn't unheard of), one of these may very well eliminate that.
The type of music you listen to also can make it matter/not matter. (If it's newer big-label, studio-overcompressed tracks, you'll never notice.)