You make this almost too easy... ;-)

"Could it be the sound card?"

Yes.

"DO all desktops support new and good sound cards?"

Yes.

You may also have low-impedance (e.g., "8 ohm") headphones that may be loading your desktop soundcard (although it seems odd that a laptop would be strong enough to drive them, if this is so). You might try a pair of high-impedance (500 ohm, 50Kohm or higher) and see if that helps.

ADD: "do the sound quality depend on the sound card quality as well?"

My own "standards" for sound are probably rather plebian. As long as it's clear and of sufficient volume, I'm happy. But somebody with an "ear" for fine music might scream in agony at my setup. ;-)

Now "clear" does still mean "NO BUZZ!" to me. Buzz is intolerable, and I don't.

Having said all that, however, I've gotten good sound out of a wide array of cards from $10 "taiwan joe" cards to $65 Sound Blasters. OTOH, be advised that I always plug their outputs into half-decent amplified speakers. I rarely use headphones, so I can't actually vouch for that functionality.

In the end, I think I've said a whole lot about nothing. So let me say something that borders on definitive:

I can infer from your description that your sound card is old, and may not be of decent quality. If such a card had "buzz", I would certainly suspect the card and replace it out of hand.

put your Electronic telecommunication device away from your computer such as hand phone or magnetic hardware system.

ctrl+AIT+Delete to find out the program that you don't need.turn it off.