1. Martha is more educated; she was a medical student before she went into the TARDIS. Rose didn't take her A-levels, if I remember correctly--anybody have a citation for that?
2. Rose became an active character when she met the Doctor. Before then, she was essentially in a rut: for example, working at a department store but not looking elsewhere for advancement, going on dates with Mickey while appearing to have no future plans with him. Part of this, I think, is related to Rose being younger than Martha. Rose hadn't made a plan for her future yet. Her experiences with The Doctor form her future for her. Martha was already actively engaged in her life when she met the Doctor, through attending medical school and, so far as we've seen, acting as mediator and advisor for her entire family.
3. Rose would talk back to the Doctor, argue with him, criticize him, and I see Martha doing the same; except, with Martha, it seems somehow less brash. Martha's self-confidence is quieter, and she's interacting with the Doctor on more of an adult level (her initial romantic interest in him seems, to me, to be a "mature" interest, somewhat wary, instead of a crush). In my opinion, Rose's initial feelings for the Doctor were very crushlike. Her feelings deepened and became more real over time, but retained an element of fantasy, that they'd be together forever, even having been confronted by evidence that no companion had ever stayed forever before.
4. Rose's support system, at least what we saw, consisted solely of her mother and Mickey, with occasional mention of a female friend. Martha's family is larger, and it would seem likely she has friends among the medical students as well. Will this provide more stability for her in the future, or less, because she has more to lose?