I played both styles (kill very few, and kill everyone) and I was quite disappointed at how few changes there were on the Granny Rags storyline.
Firstly, if you kill him before completing his mission, it looks like there's no way to get into the Captain's Chair Hotel, which contains a bone rune. I was trying not to waste time, so I just killed him as soon as I saw him and missed out on the rune. Oh well, no biggie.
Next, you get interesting some new dialog from the Bottle Street Gang member who survived Granny Rags' attack, saying "we had no chance without Slackjaw with us".
When you finally get to Granny Rags, she has Slackjaw's body in her kitchen, still crispy from being cooked in the furnace. I don't know how she recovered his corpse, as I had the skill that turns corpses into ash when I killed Slackjaw. She tells you that you may take his belongings, but to leave the bones for her.
Slackjaw's belongings consist of some coins and the key you require to proceed. If you try to pick up Slackjaw's corpse, Granny Rags attacks you. The rune in the room after Granny Rags' kitchen is still there for the stealing.
I don't know if killing Granny Rags has any effect on the rest of the game - in my stealth playthrough, I used a sleep dart on her but the end-of-level actions list only said that I helped Slackjaw escape. In my psychopath playthrough, I killed her, but I don't think it was mentioned on the end-of-level list.