When you propose something, you have a detailed plan, all the research had been done before hand before suggesting it. A suggestion merely something comes straight out of your mouth/mind without any thought or consideration. He proposes to her... He asks for her hand in marriage. He had been thinking and planning how he can marry her. You don't say, "He suggests to her" - it would have a totally different meaning. I hope you now understand the difference in meaning with these two sentences. He suggests having dinner in a restaurant tonight. He proposes having dinner in a restaurant tonight.

"Propose" is more formal and more suitable for a business conversation. Otherwise, the difference does not amount to much.