Sifri (to Num. 6:23) says that the Torah's expression אמור להם means that they can't start "until [the chazzan] says to them." Rambam (Hil. Birkas Kohanim 14:3) understands this to mean that he has to prompt them word by word.
The underlying reason, it seems from Shulchan Aruch (Orach Chaim 128:13) and commentaries, is so that the kohanim don't get confused and say the wrong words. (For this reason, the Mechaber there writes that they can start the word יברכך without being prompted, since they're not going to get confused at that point; that indeed is the Sephardic custom.) That is probably less of a concern with the berachah, since in general the failure to say a berachah correctly doesn't invalidate the mitzvah; by contrast, with birkas kohanim itself the Torah is particular about the wording, including the fact that it has to be in Hebrew (ibid. 128:14).