"Take effect" means "begin to have an effect." In the case of medication, saying "the medication does not take effect" would mean "the medication never starts working." Doctors often tell their patients something like "This medication will begin to take effect thirty minutes after the first dose."
"Effective" means the medication causes the desired effect (at any point in time). The key difference is that "take effect" indicates the beginning or onset of an effect, while "effective" refers to the overall impact of the medication. Thus, the following conversation is very possible:
"Has that Tylenol taken effect yet?" "Yes, my headache is gone! Tylenol is very effective!"
Hope that helps. ☺️