Sound cards are technologically superior to almost any onboard sound you can get, with the exception of some ultra high end motherboards.
Depending on the sound card, they can be better at different things. The Essence STX is good for headphones, because it can push impedance of up to 600 ohms. This means you are getting well over $1,000 headphone range before you can't run your headphone off of that sound card. With onbooard sound, they can generally only push around 32 - 80 ohms of impedance, which is what mid range headphones generally have.
Not to mention signal to noise ratio. The Essence STX (this is what I will use for my examples because it is the one I own and know the most about) has 124 dB SNR. Every 3 dB is a doubling of sound quality and crispness. I am not totally positive on the specification of most onboard audio for SNR, however I would assume it is in the 100-110 dB range. This means that sound quality difference between my STX and onboard is 4-8x better.
Not to mention the fact that usually onboard does not have input for things like 7.1 surround. It also does not usually come with the suite of software that people may like for their audio equipment, such as Dolby Headphone.
Now, if you have $50 headphones or some cheap speakers, will having a nice sound card make a difference? Probably not, because the bottleneck in that setup is probably your equipment, not the actual source of the sound. However, if you have really nice speakers or headphones, you would most certainly notice a difference between onboard and dedicated. Most notably would be the background static.
Also, just as an example, my Beyerdynamic DT770's have a technology in them that allows them to amplify lower noise signals, so I can pick up bass that would otherwise not be heard. If I were to plug these into onboard sound, the source of the sound itself will not pick up this low frequency, making that technology in my headphones useless.
tl;dr: Dedicated sound is better in every way, shape and form to onboard sound as long as your actual sound equipment is not the bottleneck.