Answer: Both tan and tanto (the latter of which can exist in feminine and plural forms) can be used in comparisons such as "as ______ as," which becomes either tan _____ como or tanto _____ como. One way to think of the main difference, as you suggest in your question, is that tan is used as an adverb in those constructions, while tanto is used as an adjective.
Tan basically means "so," sometimes "such a" or "as," and is used only before adjectives or adverbs (or nouns used as adjectives). Examples: Rita es tan alta como María. (Rita is as tall as María.) Rita habla tan rápido como María. (Rita talks as fast as María.) Sentences like those are by far the most common usage of tan.
Tanto basically means "so much" or "so many" or, when used with como, "as much" or "as many." Examples: Tengo tanto dinero como Juan. (I have as much money as Juan.) Tengo tanto dinero que no sé qué hacer con él. (I have so much money I don't know what to do with it.)
Tanto also can be used to make other kinds of comparisons and has a wide variety of colloquial uses; under some circumstances it can be used not only as an adjective but also as a noun, pronoun or adverb. A good dictionary will list at least two dozen different uses. Examples: Tengo quince y tantos nietos. (I have 15 or so grandchildren.) No quiero estudiar tanto. (I don't want to study so much.) No como tanto ella. (I don't eat as much as she.)
Note: This question and answer were adapted from a discussion on the bulletin board.