「[男]{おとこ}の[顔]{かお}になったな。」 could mean two very different things.
1. "You/Someone now look(s) like a full-fledged adult."
This is the meaning for which the phrase would be used more often than the other I shall explain below. It is a well-intended compliment and the speaker has been waiting for this to happen -- most likely for years.
Even though the word 「顔」 is used, it does not mean that only the person's face has changed. It includes how he acts, speaks, etc. It is talking about his whole person that has grown to reach the level where he could be called an adult.
One thing that I feel is important to mention as a Japanese-speaker is that unlike what you said in your question, this phrase is not really something a father would say to his son. It would be more like what an uncle would say to his nephew, a male family friend to the son of that family, etc. There might be exceptions where a father might say it to his son, but it would be rare in Japanese culture if not in yours.
2. "You/Someone now look(s) like a hungry wolf."
This implies that even though someone was looking like a nice husband/father/whatever-his-profession-is only 30 minutes ago, but now that he is out on the street at night only after one thing. "Two faces of a man" is what one is talking about.