Best Answer - Chosen by Voting
haru1ban is right.
The particles 'ka' and 'ne' are different. With 'ne' the person asking already has a general idea that the other person doesn't have much time, so it's more seeking confirmation. "You don't have much time, do you?"
When a verb ends with 'ka' it is just a question. "Do you not have much time?"
However, when it comes after the negative form of certain verbs, it can also imply an invitation to join the speaker.
"Suwarimasenka?" -Won't you sit down?
"Tabemasenka?" -Won't you eat?
I'm not sure if this works with "arimasen" as well, because I've never heard it used like that before, but someone else will probably know.
Best of luck. :)
