wager vs bet on

wager is a noun but bet on is not a noun.

Word	Noun	Adjective	Verb	Adverb
wager	Yes	No	Yes	No
bet on	No	No	Yes	No

As verbs, bet on is a hyponym of wager; that is, bet on is a word with a more specific, narrower meaning than wager:

Other hyponyms of wager include back, gage, game, punt, stake, play, raise, see.

wager (noun)	bet on (noun)
the act of gambling
the money risked on a gamble

wager (verb)	bet on (verb)
maintain with or as if with a bet	place a bet on
stake on the outcome of an issue