The Macintosh print technology, especially with Adobe applications, relies on a Postscript RIP (Raster Image Processor). What a RIP does is to take data and construct a raster-based image from that data. A Postscript rip is there to specifically interpret postscript data to a raster image for output.
Adobe applications on the Macintosh send data as postscript to be printed. If the printer being used does not have a Postscript RIP then the printer interprets the data as best it can. This often results in prints which are based on the embedded, low resolution, preview image within the postscript data, not the postscript data itself. So you get essentially a "this is the best I can figure out" type of print. Often vector data is inaccurate, or just looks pixelated. The printer kind of spits out a "fax-like" sheet rather than a constructed image from the mathematical plotting postscript data provides.
Illustrator especially looks for Postscript Level 3 support. There are various support level for Postscript - level 1 , 2 and 3. Similar to "versions" but called "levels". The higher the level the better the interpretation for more advanced Postscript.
You can print to a Postscript level 1 or 2 device. However you might find some aspects of Illustrator artwork look poor (such as gradients) while others appear fine. Essentially it's all a "crap-shoot" without Postscript level 3 support. You get what you get and it's not about how you've set things up or any "problem" with Illustrator. It's a limitation of the printer itself and it's ability to support some technologies.
To circumvent the fact that a printer does not support Postscript Level 3, you can save the Illustrator file as a PDF, then print the PDF.
Why this works...
Acrobat (PDF/Distiller) is a software Postscript RIP What Acrobat/Distiller do is read the postscript data and then construct a high quality image based upon the postscript data itself. The exact same thing a Postscript Level 3 printer would do. The difference is only that Acrobat/Distiller do it all within the software and don't spit out a page. Instead, Acrobat embeds a new (high quality) preview image based upon the postscript data, ignoring the existing, embedded, low resolution, postscript preview.
So, if you print from Acrobat/Reader you are printing a high resolution raster image Acrobat created directly and not relying on the printer to interpret postscript data or to print the low resolution embedded preview image.
Why Windows does this better for you....
Windows doesn't use Postscript. Windows uses PCL (Printer Command Language). So postscript support on a Windows machine isn't anywhere near as imperative as it is on an Apple machine.
Many lower-end, less expensive printer will support PCL. In fact it's hard to find any printer which doesn't. However, most lower-end printers do not support Postscript. And if they happen to support postscript, level 3 is yet another matter. This is why the same printer can print the same thing and it appears different coming from a Macintosh compared to coming from a Windows system. PCL vs Postscript.
Inkjet printers are most notably lacking postscript support in most cases. Only the top-of-the-line Inkjet printers tend to have an onboard Postscript RIP. However some manufacturers, such as Epson, provide the option to add a postscript RIP when purchasing a printer - like buying a car with the option of having air conditioning. There are some lower-end laser printers which will also fail to support Postscript. It's not just inkjets.
Checking a printer specifications for Postscript Level 3 Support before purchase is one of those things that should be mandatory if you print from or work on Apple computers, especially if you intend to print from Illustrator or InDesign directly. Photoshop prints high quality raster images (even if you used vector tools). Basically, Photoshop has it's own internal RIP when printing. So postscript support isn't really a concern for printing from Photoshop.