Edges are jagged when printing exported PDF from Illustrator

"does your printer have postscript"

Actually, any laser printer-style machine, be it Postscript or PCL non-PS will do halftone dots, as that is the only way they can build tones.  Printing from Acrobat doesn't chnage this fact.

"The top is how I want it to look"

My guess is they printed it on their inkjet printer with a high density scatter dot, e.g. an EPSON at 1440/2880 dots. Since there are no halftone dots like one would get on a professional RIP, it would look much smoother. The issue here is that any professional printer or copy place will NOT be using an inkjet to print professional business cards, they would be printed on a high end digital printer or offest press, and these are halftone dot-based. That being said, they can up the LPI to something that works better... it's just easier for them to calibrate and run consistent large jobs through at 150lpi, but that's pretty crude these days. Unfortunately, one of the big online printers (won't say their name) prints all their stuff at 150lpi, so that was the end of them for me, as that's not good enough.

One copy shop I deal with can easily set their Xerox IGen to 175 or 200 if I ask for it.  If the OPs printer cannot do a higher line screen, THEN they should look for another vendor.

Yes, the answer is to print a spot ink, but my guess that is beyond the budget of the OP. In any case, their bottom image looks like it was printed in greyscale/black and not in colour at all: the dot pattern looks more like it's printing a dark grey and there are no other ink dot colours as there should be. e.g.:

This is a 6pt line halftoned to 175lpi.

It was supposed to be brown.

printed on cream laid paper.

It's a rectangle shape in Ai.

I am using Save As.

Adobe saving preset: [Press Quality]

About the printer/app.

I don't know what printer it is, those are business cards I am printing in a local printing shop. The printer is literally bed-sized.