What is the difference between metonymy and synecdoche?

In practice, there isn't much difference: you could arguably pick just one of the terms and use it to describe both types of rhetorical substitution. (I like metonymy: it's easier to spell, more spelling checkers know it, and the meaning is more apparent to me: meta+name.)

The difference, to the extent that it exists at all, is whether the attribute that is substituting for the whole is part of the whole (synecdoche), or merely associated with it (metonymy). So "suits" instead of "officials" is metonymy (officials wear suits, but last I checked, the clothing is not permanently attached to their skin), while "hands" for "workmen" is synecdoche.

You could also make a case for using metonymy for any example where a smaller part or attribute substitutes for a larger part or attribute, and reserving synecdoche for examples where the larger stands for the smaller, or the container for the contents. Under this interpretation, both "suits" and "hands" could be considered metonyms, and synecdoche would be something like "General Motors announced cutbacks" — it was presumably a spokesperson who issued such a statement on behalf of the CEO or board of directors, since General Motors is not a single entity and does not possess speech capabilities.

Metonymy is the substitution of the name of an attribute or adjunct for that of the thing meant; synecdoche is a figure of speech in which a part is made to represent the whole or vice versa.

Maybe now the suits in Washington will listen. [metonymy]
I liked to watch the track. [metonymy]
Cleveland won by six runs. [synecdoche]

The Grammarphobia Blog's May 7th, 2009 article may assist:

“Synecdoche” and “metonymy” are figures of speech in which one thing is used to represent another. In both of these rhetorical figures, the original term and the substitute are closely identified or associated with each other.

In this respect, “synecdoche” and “metonymy” are different from “metaphor,” in which the terms are unrelated yet imaginatively similar (as when you call your ’67 Pontiac “a boat”).

With, “synecdoche,” a part is used to represent the whole or vice versa. Examples commonly cited are the use of “hand” to mean a sailor and “the cavalry” to mean a single trooper. It’s pronounced sin-EK-duh-kee and ^[♦]comes from a Greek word meaning “to take with something else.” (“Schenectady,” the ninth-largest city in New York State, is pronounced skuh-NEK-tuh-dee.)

With “metonymy,” the substituted word is not a part (or an extension) of the original but something associated with it. Classic examples are “the crown” to represent the monarchy and “the sword” to represent military power. It’s pronounced met-ON-uh-mee and comes from a Greek word meaning “change of name.”

Here’s a simple illustration of the difference. A new guy at the office might be described as “a new face” (synecdoche) or as “a new suit” (metonymy).

^♦If my following conjecture of the etymology of 'synecdoche' is false, then please correct me:

[Etymonline] [...]
from synekdekhesthai "supply a thought or word; take with something else, join in receiving,"
from syn- "with" (see syn-) + ek "out" (see ex-)
+ dekhesthai "to receive," related to dokein "seem good" (see decent). [...]

dekhesthai combines with syn- to mean: 'receive with'.

If interpreted metaphorically beyond "out", then ek means not just 'out', but 'something else' because 'something else' must necessarily be 'out' of the scope of whatever 'something' is.

In toto, dekhesthai + syn- + ek = 'receive' + 'with' + 'something else'.