Remember is when a memory is involuntary and memorize is an active project you (consciously and deliberately) decide to undertake. You might remember that you had something to do, but you sit down and memorize a list of vocabulary in order to pass your language test on Friday, for instance.

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To remember is the process of recalling information. To memorize is the intentional process of dedicating, or attempting to dedicate, information to memory. They're separate processes involving the same thing. Like recording a movie and playing a movie.