From the Doctor Who episode Journey's End transcript:
DONNA: I thought we could try the planet Felspoon. Just because. What a good name, Felspoon. Apparently, it's got mountains that sway in the breeze. Mountains that move. Can you imagine?
DOCTOR: And how do you know that?
DONNA: Because it's in your head. And if it's in your head, it's in mine.
DOCTOR: And how does that feel?
DONNA: Brilliant! Fantastic! Molto bene! Great big universe, packed into my brain. You know you could fix that chameleon circuit if you just tried hotbinding the fragment links and superseding the binary, binary, binary, binary, binary, binary, binary, binary, binary, binary, binary, binary, binary, binary. (gasp) I'm fine. Nah, never mind Felspoon. You know who I'd like to meet? Charlie Chaplin. I bet he's great, Charlie Chaplin. Shall we do that? Shall we go and see Charlie Chaplin? Shall we? Charlie Chaplin? Charlie Chester. Charlie Brown. No, he's fiction. Friction, fiction, fixing, mixing, Rickston, Brixton.
(This time it hurts.)
DONNA: Oh, my God.
DOCTOR: Do you know what's happening?
DOCTOR: There's never been a human Time Lord metacrisis before now. And you know why.
DONNA: Because there can't be. I want to stay.
DOCTOR: Look at me. Donna, look at me.
DONNA: I was going to be with you forever.
DOCTOR: I know.
DONNA: The rest of my life, travelling in the Tardis. The Doctor Donna. No. Oh my god. I can't go back. Don't make me go back. Doctor, please, please don't make me go back.
DOCTOR: Donna. Oh, Donna Noble. I am so sorry. But we had the best of times.
DOCTOR: The best. Goodbye.
DONNA: No, no, no. Please. Please. No. No.
(The Doctor mind melds with Donna and takes her memories away.)
(Donna passes out.)
DOCTOR: She took my mind into her own head. But that's a Time Lord consciousness. All that knowledge, it was killing her.
WILF: But she'll get better now?
DOCTOR: I had to wipe her mind completely. Every trace of me, or the Tardis, anything we did together, anywhere we went, had to go.
WILF: All those wonderful things she did.
DOCTOR: I know. But that version of Donna is dead. Because if she remembers, just for a second, she'll burn up. You can never tell her. You can't mention me or any of it for the rest of her life.
So the Doctor-Donna couldn’t deal with having the Doctor’s mind in her brain.
But the human Doctor born out of meta-crisis earlier in the same episode never showed any kind of insanity till the end, and The Doctor also wasn't worried about that. He was talking about spending his life with Rose.
The meta-crisis Doctor had human brain, and he possessed a Time Lord consciousness with the entire knowledge of The Doctor.
Why wasn't he affected in the same, or any, way?
My mistake that I didn't realize meta-crisis Doctor was a hybrid, but Doctor-Donna was also a hybrid. In the episode The End of Time, The Master transformed all humans into himself, but Doctor-Donna wasn't affected.
To prove that Donna was really a hybrid, here's quote from Journey's End:
DALEKS: Exterminate. Exterminate. Exterminate.
(Donna works more controls on the panel.)
DALEK: Weapons non-functional.
DONNA: Phwor. Macrotransmission of a K-filter wavelength blocking Dalek weaponry in a self-replicating energy blindfold matrix.
DOCTOR: How did you work that out? You're
NEW DOCTOR: Time Lord. Part Time Lord.
DONNA: Part human. Oh, yes. That was a two-way biological metacrisis. Half Doctor, half Donna.
DOCTOR: The Doctor Donna. Just like the Ood said, remember? They saw it coming. The Doctor Donna.