The correct distinction between synecdoche and metonymy is:
Synecdoche uses a part to refer to the whole; metonymy uses a related idea to refer to something closely associated with it.
To elaborate:
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Synecdoche involves using a part of something to represent the whole (e.g., "all hands on deck" where "hands" refers to sailors).
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Metonymy entails substituting the name of one thing with the name of something else that is closely related or associated (e.g., "the White House issued a statement," where "the White House" refers to the U.S. President or the administration).