Dumbledore and Harry don't actually do anything very different from each other in that scene. They're both blaming themselves instead of McGonagall. What's different is in how they express that. Harry is very clear about who he is blaming, and why; he tells McGonagall exactly what she did wrong when she asks to be blamed, although he still does not in fact blame her. Dumbledore, on the other hand, offers only comfort; he doesn't even tell McGonagall that he's going to blame himself, although she can very well guess.

It's also worth noting that Harry chooses an interesting fault to explain to Professor McGonagall. He doesn't suggest that, like Quirrell, she should have checked on all the high-value targets before leaving the room. Instead, he told her to trust her students more. This is something that McGonagall could actually do; it's much better suited to her than the more complicated, strategic options Harry would suggest for himself or Quirrell. So, although it's definitely not explicitly said this way, it's pretty easy to read Harry as giving advice here, which Dumbledore notably fails to do.