This was a ridiculous and sloppy scene where the exposition and mythos of the film utterly failed, particularly Spiderman forgetting he could stick to walls but also Curt Connors had no reason to save Spiderman given what we're shown about his character, and no I'm not observing Kurt Connors as the character from the comics because this is an adaptation (film version only). In short, Kurt Connors saved Peter Parker not out of some respect from the twerp but because of hack writing to create shock value.
This whole scene was a faux pa, the only reason it was included was because of bad writing: Captain Stacey had already been slain and Spiderman forgot he was part spider for some reason so only Kurt Connors could save him? What?
If you believe this is an outlandish assumption ask yourself if Captain Stacey hadn't been fatally injured would he have saved Spiderman? The answer of course is yes but because he'd been removed from the picture only Kurt was left to save Peter. You see there are other scenes where sloppy writing has resulted in swapped character traits. For Example, in Star Wars Episode 2 when Senator Amidala was about to be killed, Obi-Wan rather than Anikan jumps after the probe.
Since I personally believe the easiest explanation is the most likely, it's more than likely and also proven the creative writing in this movie was severely lacking, see any movie review online for reasons why - it's also more so likely Captain Stacey was killed prematurely b/c of what we already know about Gwen Stacey - SPOILERS!!!
She's killed by Green Goblin in the comics!!! Yep, she dies from a broken neck.
Since a sequel is already on the horizon and Green Goblin is confirmed as one of the main baddies in the trailer as well as the main heroine of this film and the sequel being Gwen Stacey it's possible the writers in order to create dramatic effect likely had her father killed to increase the levity of her death, resulting in Spiderman's two biggest failures. This resulted in Connors briefly swapping character traits to ensure Peter lived to see another day . . . even when it was completely unnecessary.