There's still some lingering mild debate out there in the bowels of the internet on whether imply and infer are synonyms or antonyms, but I think it's safe to say that the antonym camp has the numbers and so, for clarity, allow me to explain to anyone who still doesn't quite get it what the difference is and when you want to use each one.
Simply put:
The speaker implies.
The listener infers.
Arguing that implying and inferring are the same thing and can be used interchangeably is like arguing that pitching and catching is the same thing. There's clearly a relationship there. Someone is implying and someone is inferring but it really helps if you understand the difference. Talking with someone is a lot like playing catch but when relating that conversation to others it helps to know who is pitching and who is catching.
Small thing. But the better we learn to communicate the more persuasive we'll be. And that seems important to me.
Imply is to suggest a certain condition or to indicate that something is possible without direct delineation. If someone were to say, "Give me your money, or else", they are implying that they will injure you in some fashion if you do not give them your money. If that same person were to say, "I shot the last guy that didn't give me his money", the inference would be that he will shoot you too, if you do not give him your money. The second statement isn't quite an implication, it is more of an inference. I believe the difference is that an inference is based on a fact that would lead you to conclude a logical outcome based on correlation.