Well, the first thing to notice is that, at this point, Harry has a lot more experience with battling Dementors than Hermione, because of his practice sessions with Lupin. True, he'd been taking on a boggart (and only one), but Hermione has only ever encountered the Dementor on the train and that one wasn't especially hostile. Harry, however, has built up a bit of resistance and he also knows what to do - albeit he does fail to do it.

More importantly, though, Harry's just been offered a home with Sirius and a way out from the Dursleys:

>'I'll understand, of course, if you want to stay with your aunt and uncle,' said Sirius. 'But ... well ... think about it. Once my name's cleared ... if you wanted a ... a different home ...'
>
>Some sort of explosion took place in the pit of Harry's stomach.
>
>'What - live with you?' he said, accidentally cracking his head on a bit of rock protruding from the ceiling. 'Leave the Dursleys?'
>
>'Of course, I thought you wouldn't want to,' said Sirius quickly. 'I understand. I just thought I'd -'
>
>'Are you mad?' said Harry, his voice easily as croaky as Sirius'. 'Of course I want to leave the Dursleys! Have you got a house? When can I move in?'
>
>*Harry Potter and the Prisoner of Azkaban* - pp.277-8 - Bloomsbury - chapter 20, *The Dementors' Kiss*

You can bet your bottom dollar he's gonna fight with everything he has for his godfather and his potential new life. And we know that, when the chips are down and push comes to shove, Harry really can be the hero. He also has hope - more hope than perhaps he's had in a long time. A powerful weapon against the Dementors.