I don't know that it really gives you a great sense of what the show is like considering the Doctor is barely in it. But still, it is an excellent episode and stands on its own very well.
I'm not a huge fan of "Rose" but I think it does do a good job of establishing who the Doctor is and a lot of the other stuff that's going on.
I've always said that "The End of the World" is a good Doctor Who litmus test. If you like it, you'll probably be a loyal fan of the series. If you don't, then you'll still like the show but should probably confine yourself to a narrower, higher quality bracket of episodes.
Miraculously, I've managed to get my mother hooked onto the show with a smattering of high quality episodes, frequently ones that have a strong emotional or comedy thread to them. I realize she'll never have the patience for something like "The End of the World" or "The Impossible Planet"/"The Satan Pit." But there are a bunch of other episodes that I have exposed her to quite successfully:
"Rose"
"The Unquiet Dead"
"Father's Day"
"The Shakespeare Code"
"Blink"
"The Unicorn & the Wasp"
"Silence in the Library"/"Forest of the Dead"
And then a bunch of the Matt Smith episodes, which have a stronger vein of comedy than the Eccleston/Tennant episodes and don't sink into angst quite as often. I realize you said no Season 5, but I still highly recommend:
"The Eleventh Hour"
"The Vampires of Venice"
"Amy's Choice"
"Vincent & the Doctor"
"The Lodger"
Hell, maybe you're best off just hitting all of Moffatt's episodes from Seasons 1-4:
"The Empty Child"/"The Doctor Dances"
"The Girl in the Fireplace"
"Blink"
"Silence in the Library"/"Forest of the Dead."
