They have the same basic meaning. However, when we use the singular form, it usually means that this is the order/philosophy/command/rules/promise, etc., of the speaker. So we might say, "In my house, my father's word is law!" We say, "I give you my word" to mean "I promise"; to say, "His word is gold" means you can trust him/believe him.
To talk about a person's "words" (using the plural form) means simply to repeat what they said... the words aren't especially important or dramatic.