Yep you will see a difference because common printers have even less of a colour gamut. I've done tests being very careful to control all the variables every step of the way from the original raw to the printer, even doing one over saturating first to intentionally blow the colours out of gamut range. I can say I have convinced myself it is no where near worth the effort. Not even close.
Converting the image to sRGB and printing with auto colour management ticked produced nearly identical results to converting to PhotoPro or Adobe RGB, setting up my printer proof with the colour profile for the paper, setting up adobe colour management, and then disabling colour management in the printer driver.
Ok admittedly I am sure if you take it to a professional lab that handles the format you may actually get better results, or if you have one of those 8 colour Epson printers too. But with a standard 5 colour photo printer all I established is that I wasted an afternoon trying it out. And it definitely is not worth the effort to me to archive in anything other than sRGB since I then need to convert it everytime I do something basic like upload it to flickr or send it to a friend etc.