Both of them seem to be able to be used to mean instruct. Am I right in saying that you can 嘱咐 and 吩咐 someone to study hard? If so, what is the difference in meaning between these two terms?
On the most basic level they both mean "to tell", but they are not exactly the same - there is some nuanced differences.
吩咐is closer to English "to instruct". It is often used to describe a superior, or elder, telling their juniors to do something. Thus, it carries the connotation of a verbal command, almost ordering someone to do something. But it is less forceful outright than 命令, "to command".
嘱咐in contrast is more even, and thus closer to English "to ask" or "to urge". It is also used to convey concern, although it doesn't necessary carry any connotations of concern by itself.
Despite these nuances, for the most part the meanings are similar enough that they can be used interchangeably. That's why you'll also find them listed as synonyms.
他爸媽吩咐他要照顧好妹妹: His parents told him to take good care of his sister. Alternatively: "instructed him", or "ordered him".
他爸媽嘱咐他要照顧好妹妹: His parents bade him take good care of his sister. Alternatively: "asked him", or "urged him".
Note that apart from the meaning of "to tell", 嘱咐 has a second meaning "to entrust", or "to charge".