You would not be able to acquire two gestures:
Extreme Repentance is the first one and requires you to win the first Patches fight, initiate another and then surrender. Patches' Crouch is the second one and it's only available when you complete his entire quest line.
You would not be able to complete one of the quests in
which awards you with a weapon and an armor set (which is somewhat interesting option for heavy load / high poise builds) :
Magma Whip Candlestick and Bull Goat Set
That one Patches quest is an optional part of the larger quest line, you still will be able to complete it and get the final reward.
Of cause you will also miss multiple conversations/interactions with Patches, like his typical
"push you down to some hole" thing in Mount Gelmir
and another NPC interaction:
dialogue with Tanith after giving her Dancer's Catanetes