Imply - meaning is suggested from words that are not explicitly said. The person wants to suggest meaning beyond what he or she actually says.
To imply - you must be the speaker, writer, or sender.
Infer - a conclusion is drawn or deduced from words that are not explicitly said. The person reads meaning beyond what was actually said in someone's message.
To infer - you must be the listener, reader, or receiver.
My brother's boss bought him a watch for Christmas. I think his boss was suggesting that he start arriving on time for work. (In this example, the boss IMPLIED that my brother needs to keep better track of time)
When trying on a pair of jeans, I asked my friend what she thought. She told me she thought they were too tight. I was offended because I thought she was really telling me that I was too fat. (In this example, I read into what my friend was saying. I INFERRED more meaning into what my friend said than what was intended)