Ultimately, this is a "different strokes for different folks" kinda thing. I personally can't stand Fallon, but I can see why he's so popular.
If you want more ammo for future fights with your friend, you might want to consider this:
There are two types of comedians - clowns and talkers. Clowns rely mostly on their performance to get laughs. Talkers, obviously, rely mostly on their words, their writing. Sometimes there are comedians who excel as both clowns and talkers, but for the most part comics fall more into one camp or the other.
Fallon is a clown. Notice how his monologues tend to be pretty short; that's because I don't think he nor his staff are particularly good comedy writers. Look at the Tonight Show Youtube playlist, and you'll see that his monologues get the least views. Watch them, and you'll see that he doesn't get many laughs. Because the writing is a weak spot, he has to make up for it with outrageous performances. He'll dress up in drag. He'll cake on makeup to look like Trump. He'll play elementary school instruments with the Roots and Metallica. Etc.
Don't get me wrong, Fallon is a good clown. His performances are high-energy, he's a talented impressionist, and his musical segments are thoroughly entertaining. But his jokes, in my opinion, aren't very funny. His interview style is also pretty obnoxious. That's why he's not known for hilarious monologues or insightful interviews. Instead, he's known for playing fun, silly games with celebrities. If you like that kind of stuff, then great.
Colbert is a talker. He has very good comedy writers on his staff. As for Colbert himself, he's a surprisingly intelligent person with both a talent and a lot of training in improv. I think Steve Carell once described Colbert as "the smartest man I know." You can see what he means whenever Colbert talks.
Colbert isn't that good of a clown; at least he isn't as good as Fallon. He isn't that good at impressions. He doesn't often dress up in silly costumes. But his monologues are top-notch. That's why when you go to to the Late Show Youtube playlist, you'll see a lot of monologue videos - some of which are around 10 minutes long. Plus you'll see that his monologue videos tend to get the most views out of anything else posted on the channel.
Fallon is a clown, Colbert is a talker. Colbert is winning in the ratings right now because people are freaking out about the political situation, and they want comedians who'll make them feel better about it. That's a job for a talker, not a clown. On the flip side, Fallon dominated the ratings when the country was pretty chill about politics, and wanted to see a late show host and Emma Stone throw buckets of chicken through a basketball hoop rather than hear jokes about the president.
Whether someone's a clown or a talker, there are situations that they're best suited for, and there will be people who prefer one form of comedy over the other.
