That was one thing about Web of Death which bothered me. We never got to see his actual reaction to Peter being the one under the mask.
Part 1 of Web of Death ended with Ock taking off Peter's mask. Then at the start of part 2 he was busying himself making the cure. Thinking about how easy it would be to kill Spider-Man there and now, as he is completely at his mercy. But he'd rather do it with Spidey in his prime. Kill him in "the dance' as he called it.
I think the reason he didn't freak out is because the unmasking was under his conditions. He unmasked Peter, and the secret was his to know. Peter was also dying.
In Civil War, Spider-Man goes and unmasks himself to the world, telling everyone he's been doing this since he was 15 years old.
It's like a slap in the face to all his foes who he beat back then. "Yeah that's right, you got your asses whupped by a 15 year old kid". No wonder he flipped out. Also, since Aunt May knew, Ock also assumed she knew back then too. And she and Peter had been laughing at him behind his back.
So I can understand why he flipped out in Civil War, and not in Web of Death.