The issue most likely starts with you being in one color working space in Quark and another color working space in Illustrator.
I'll assume you're probably working in CMYK, and what that means is that to get to the equivalent of a PMS color in each color space, you're going to have differing CMYK values.
If you're in RGB the same rules basically apply, with an even further twist thrown in by Illustrator.
If you're printing to a RIP, and the RIP has a PMS library and you have it on and operating correctly, then it will account for input color space discrepancies and look for the closest match it can get to the L*a*b* value of the called PMS color in the destination color space--your printer color space...
Of course in order for that to happen, you have to be using color management, and the engine you're driving the printer with has to have a PMS library. If both of those things aren't true, then not only does your printer driver (RIP or otherwise) have no idea what PMS216 is, it has no idea what color space to render it into. So in that case what it does is just takes the raw CMYK numbers the file refers to as PMS216 and passes them through to print.
If in your two programs you're creating in two color spaces, then of course those CMYK numbers will be different, and your final color will be different.
Fact is there's really no such thing as "no color managing" and every pixel in every digital file you create relates to some color space. You can choose it yourself or have your applications choose it for you. But even if you choose "no color management" all the application does is use a default working space and then not tag the file. Digital color is color by numbers. If you're going to try and do it, it's pretty simple: You can either put the numbers to work for you...or you can fight them.
And if you fight them, my experience has been that they always win.
Mike Adams
Correct Color