First, to answer your initial question, the two images will appear differently after converting to CMYK because the RGB color space, or gamut, is significantly larger than the CMYK color space. Colors in the photo that exist inside the RGB space, but fall outside the CMYK gamut will need to be brought inside the space to complete the conversion.
How colors are brought into the space is determined by Rendering Intent. If you’d like an explanation of this process, I’d be happy to explain.
In the end, like PrintDriver recommends, reach out to the printer, and see how they would like you to send the images.