In general, a suggestion is something you propose that the OTHER PERSON do. An offer is something you propose YOU YOURSELF do.
suggestion: "You look tired. Why don't you take a break?"
offer: "You look tired. How about I make you some tea?"
to suggest: "I suggested she take a break, but she wouldn't listen to me."
to offer: "I offered to make her some tea, but she didn't want any."
So, if the speaker uses "suggest," the LISTENER is responsible for the act.
If the speaker uses "offer," the SPEAKER is responsible for the act.
There are exceptions to this rule, of course. This is just a broad generalization.