This is similar to what Matt Эллен said, but uses the analysis and terminology of the Cambridge Grammar of the English Language. First, tiny is only ever an adjective. As such it can function as attributive modifier of nouns (e.g. a tiny flower) or predicative complement of verbs (e.g., that flower is tiny), and a few other minor functions.
The situation with little is similar when it's an adjective, but it's not just an adjective. We also have the determinative little (e.g., little water) and the complex determinative a little (e.g., a little water; a complex determinative is a single determinative made up of two or more words.) Determinatives typically function as determiner of nouns (e.g., that water), but some determinatives also function as modifier of adjectives (e.g., it wasn't that big). Most adjectives, tiny included, can't function as modifier of adjectives. Also, some NPs can modify adjective (e.g., [a city block] longer).
That made me [a little] happier
D a little is modifier of Adj happier
*That made me tiny happier
Adj tiny can't modify Adj happier
*That made me a tiny happier
D a can't form a constituent with Adj tiny or Adj happier; Adj tiny still can't modify Adj happier
D a is determiner of N bit forming an NP
That made me [a bit] happier
NP a bit is modifier of Adj happier
a little/tiny bit
D a is determiner of N bit; Adj little/tiny is modifier of N bit
That made me [a little/tiny bit] happier
NP a little/tiny bit is modifier of Adj happier
The interesting question is how little became a determinative and how a little became a complex determinative.