What can Illustrator do that Photoshop can’t?

I’m fairly expert in Photoshop but a total novice in Illustrator. I need to do an infographic and I have a suspicion that Illustrator is what a real designer would use, but I don’t really know why I think that, since Photoshop does vectors pretty well and I’m not sure what Illustrator adds to this. (I’m not saying it doesn’t add a lot, I just don’t really know what it is.)

Can I Put my logo on an outputted PDF?

I create designs and then export to PDF and give them to the printers, but what I would like to do is to put our company logo on there, but not in the printable area. I know I can put the file name etc around it but is there a way to put a logo too? If not, is there a way to put my company name/author name without changing the file name?