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Both games have their own advantages.
Gold and Silver are the base games.
Crystal includes plenty of additions that make GS a smoother experience, just like how Emerald or Platinum add things to fix some flaws RS and DP have. These include making contacts on your phone more useful, like Schoolboy Alan handing out Fire Stones, or Pokefan Beverly giving out Nuggets every now and then. Encounter rates have been improved drastically. You no longer have certain Johtomons be so rare, like Snubbull or Marill, and can get an early Growlithe, though it does get rid of some mons like Mareep's line, like how Emerald removes Roselia and Platinum gets rid of Murkrow. The music has a pretty different tone from the remakes' remixes, though there are some things missing, most notably, most Kanto towns use the Saffron theme rather than remixes.
HGSS introduces plenty of side content to keep players interested. These include the Pokeathlon, Pokewalker features, the Safari Zone, rematches, Frontier, and if you're playing an international version, Voltorb Flip. The encounter rates are from GS, so while Mareep exists again, it'd take you a while to get Marill and other mons. Legendaries play a bigger role, you have to fight Ho-Oh to beat the game. GB Sounds also exists if you dislike the remixes, but it shows up a bit too late and isn't all that faithful to the chiptunes.
It all depends on what you prefer. Crystal streamlines the maingame, HGSS has great side content.
