Memorize means to actively commit something to memory (in other words, to “put something into your memory”). Remember is the process of “pulling something out of your memory”. So they happen at different times.
For example:
“The night before the test I memorized the vocab words.”
“While taking the test I could not remember the meaning of the word.”
Also, ‘memorize’ is usually done by studying. But ‘remember’ doesn’t have to be something you studied.
For example you can say
“I remembered how to say __ in English”
And you can also say
“I remember when we went to the US”
But you cannot say
“I will memorize this trip to the US”