no, no, no, no, no, it's different, if you pronounce sz, you put the... dorsum (back?) of your tongue against the area between alveolae (google translator says gums???) but if you pronounce rz, you point the same plase more with the apex linguae (maybe the top of your tongue) stay calm, it's still much easier than ř in czech that is quiet similar to rz :D
"Przykro mi" is more like "I feel sorry/bad". It can convey apologies for something that you've done, and show that you feel bad about that. It can also be used when your friend tells you that his mother just died (I'm sorry to hear that). "Przepraszam" works only in the first context, plus as an "excuse me".
To tease this apart a bit more, "przykro" is an adverb that describes an experience as being grievous. "Mi" is just the dative case of "ja," meaning "me." (It's dative here because the regretful state is something you experience, not exactly something you do, so you're grammatically considered the indirect object).
"Przepraszam" is just the first-person singular conjuation of the verb "przepraszać", to apologize.
So the distinction Jellei describes is very similar to the one in English, where you might say "I'm sorry" either to express sympathy or to express fault, while "excuse me" or "pardon me" can be either an apology or just a polite attention-getter.