The only card in Duel Links that Ancient Gear Frame can search that Wyvern can't is Spell Gear, so most of the times you will Summon the latter, but being able to thin your Deck is always nice and you won't always start with Wyvern in your hand.
By discarding a card, Ancient Gear Frame allows you to add to your hand an Ancient Gear Golem or a Spell/Trap Card that specifically lists “Ancient Gear Golem” in its text.
