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The Apricorn Balls are very interesting in HGSS, since they aren't in DPP's data, there's a lot of careful attention made to keep compatibility. Firstly, unlike every other Poke Ball, they do not have a unique animation when you send out a Pokemon in them, despite that being a feature since Gen 3. The Apricorn Balls were finally given unique entry animations in Gen 5, but you can at least in Gen 4 see the ball when you send a Pokemon out. These balls can also not be held, will appear as normal Poke Balls when participating in Link Battles, or when using the VS Recorder, despite being visible in the Battle Frontier. Though despite being displayed as a normal Poke Ball when traded over to the Sinnoh games, or Battle Revolution, they will keep their native ball when traded back to HGSS or moved forward to Gen 5.
The Friend Ball, Lure Ball and Level Ball remain the same, but the rest have been tweaked in some way:
The Moon Ball has now been fixed and will have a improved 4x catch rate on all Pokemon that can evolve from a Moon Stone, including the Skitty family. Though, uh, Skitty or Delcatty cannot be encountered in the wild in HGSS, so that specific effect is never used. Since Nidoqueen, Nidoking, Cleffa, Clefable, Igglybuff and Wigglytuff also cannot be found in the wild, uh, that leaves six Pokemon that actually benefit from the effect (both Nidos, Nidorina, Nidorino, Clefairy and Jigglypuff). Great.
The Love Ball is now fixed to have a 8x catch rate if the opposing Pokemon is the same species, but opposite gender to your current Pokemon. Or they broke it horribly like what white cis males always fucking do hissssssssssssss
Uh, anyway, the Fast Ball has now been changed to a 4x catch rate of any Pokemon with at least 100 base Speed. This is pretty great, as that applies to 44 Pokemon in Gen 4, though only 14 of those can actually be caught in the wild (Dugtrio, Electabuzz, Electrode, Kadabra, Latias, Latios, Lugia, Mewtwo, Persian, Raikou, Rapidash, Rayquaza, Sneasel and Tauros) and this does leave out the other two doggies, Entei and Suicune. Regardless, it still has its uses.
Finally, we have the Heavy Ball, which has tweaked its weight requirements. Here's a comparison between the old weights and the new ones.
The only Pokemon that can now receive the new +40 boost to catch rate is Snorlax and Groudon, while Kyogre gets the +30 boost, making the ball pretty damn useful. Ironically, a lot of Gen 4 legendaries: Dialga, Regigigas, Heatran and Giratina also benefit from the +40 boost, but cannot be encountered in HGSS in the wild. Otherwise, the various rules from GSC still apply, that sometimes, it's only slightly better than an Ultra Ball, in regards to some Pokemon like Steelix. And while there are a few new Pokemon that benefit from this ball, since they can't be encountered it makes the whole thing moot.