Yes, each printing systems uses a color profile to print. Some can use several profiles, some can only use one. Best system for printing is cmyk.
Okay, let's try it with an example.
Supose you have a pdf file which has included two images. The pdf file uses for example cmyk, one image too, the other maybe rgb.
A printing house uses usually a special software for printing, for example to print a poster with a0 they will use a rip software. This software looks into the file and trys to recalculate the rgb information to cmyk, because the printing systems of print houses are build to use cmyk informations.
So your pdf file with two color models rgb and cmyk has changed after the rip software to one color model cmyk. Now the system can print it.
If you print the pdf file on your own printing system at home, you use usually something like acrobat (or sumatraPDF) to show and print the file.
The software Acrobat professional (you have to buy it) has a tool Preflight inside, which can do parts of the job of the former rip software. It shows you which color models and which color profile are used in the pdf file and allows you to change it with preflight.
Rgb colors and cmyk colors are not the same. Even if one changes RGB to CMYK there can be roundig effects so that you will get with two different prints two different color changings from rgb to cmyk.
To avoid this you should use for one pdf file only one cmyk model and for all images. Then colors are not recalculated for printing (with at last rounding effects).
Suppose you want to print a poster and be sure that you will get the color you saw on your monitor. That is only possible, if you use a callibrated printing system (very expensive!), with callibrated screens and printers. That mean you see a rgb color on the screen and it has to be changed to cmyk without changing the printed color. At last a printing system works with two calulations:
RGB -> meta color sheme -> CMYK.
Now let us have a look to the printer. Printer can use different colors (inks), for example 3 or 6 or 8 to print the wanted color. Then there is the color of the paper. White is not white, one paper needs more ink, others less. So to print color X on paper A, you need a special cmyk value, for paper B another, different cmyk value. These recalculations are included in the different cmyk profiles.
A callibrated printing system uses all theese informations to print the wished color X. The printer software (rip software), the rgb model for the monitor and the cmyk model for printer with used paper makes as sure as possible, that you get color X on paper like color X on screen.
If you print on paper A (say thin, cannot take much ink, white 1) you need a special cmyk profile to get color X. With paper B (thick, needs more ink, more time to print, photo quality) and the same printer you need another cmyk profile. Then the rgb color is recalculated to cmyk color model in the printing software (rip software) to the correct value for color X.
With cheaper systems, like a home printing system, color profiles are similar used, but the system can't be calibrated. Thats the reason why you always need one or more test prints ...
So to answer the question: yes, color changes if you change the color shemes.
Does it help now?