When defining a syntax table for a major mode, I can define both word constituents and symbol constituents.
When should I use each? For example, given a programming language that writes variables in snake case foo_bar, should _ be part of a word or symbol?
Word constituents: ‘w’
Parts of words in human languages. These are typically used in variable and command names in programs. All upper- and lower-case letters, and the digits, are typically word constituents.
Symbol constituents: ‘_’
Extra characters used in variable and command names along with word constituents. Examples include the characters ‘$&*+-_<>’ in Lisp mode, which may be part of a symbol name even though they are not part of English words. In standard C, the only non-word-constituent character that is valid in symbols is underscore " _ ".
Courtesy of gnu.org
When should I use each? For example, given a programming language that writes variables in snake case foo_bar, should _ be part of a word or symbol?
It depends one what you want to achieve. I'll say, '_' should be part of word. That way, foo_bar will be treated as one word.
Please throw an eye to that question, there is an interesting answer talking about superword-mode and subword-mode
This answer addresses your question title: "What's the difference between words and symbols". It does not speak only to the body of your question, which is about symbol syntax, which has been answered well by @Nsukami.
There are two very different meanings of the word symbol in Emacs:
symbol syntax, which involves word-constituent characters plus symbol-constituent characters. This can apply to any mode, in particular any programming mode/language. @Nsukami described this meaning quite well.
Emacs-Lisp symbols. This is not a syntax category. A Lisp symbol is a Lisp entity, or object, which has certain characteristics: a name, a plist (property list), and potentially a value and a function definition.
By default, the value cell and function cell are empty, and the plist is is the empty list (nil). Lisp functions symbol-name, symbol-value, symbol-function, and symbol-plist return the components.
Note, BTW, that there is no mention of these things in the Emacs manual; they are covered only in the Elisp manual. And more significantly, perhaps: There is no index entry that matches "symbol" for the first meaning (syntax). All index entries that include the term "symbol" refer to the second meaning (Lisp symbol). In particular, there is no entry that contains both "symbol" and "syntax". Whether there should be such an index entry, I don't know (probably). But this can give you some idea of the importance to Lisp of symbols as Lisp objects.
For more info, see:
The Elisp manual, node Symbols
The Elisp manual, node Syntax Class Table
The Glossary on Emacs Wiki.
This is explained in the ELisp manual.
Emacs uses terminology from Lisp. What Lisp and Emacs calls a symbol is what many other programming languages call an *identifier. That's the name of a variable, function, etc.
A word is a run of letters or digits with no intervening punctation.
Most programming languages allow the underscore _ character in identifiers, in addition to letters and digits. Some languages, such as Lisp, allow more. In programming languages, characters such as _ that can appear in identifiers have the class “symbol constituent” _; letters and digits have the class “word constituent” w.
Commands that act on words treat a sequence of word constituents as a word. Commands that act on expressions (e.g. forward-sexp) move by whole identifiers at once. Syntax highlighting usually treats all symbol constituents identically.
