FWIW, there was a Rambo reference in A Fish Called Wanda that always made me laugh:
Two actually. One at the top of the steps, and one inside as an exhibit. Though that’s temporary thankfully.
Don’t forget, Rocky beat Thunderlips in what was supposed to be a friendly exhibition between a boxer & a wrestler but turned into a no-holds barred bloodbath.
Never mind. 
Like the Taliban?
True, maybe I’m selling Rocky a bit short.
But, if any type of weapon or improvised man-trap enters the equation, that would surely be all Rambo.
I think the Rocky franchise is more accessible. Family-friendly.
I never saw any of the Rambo movies. I was a kid when the first ones came out, so I never saw them. Then when I was old enough to see them, I wasn’t in to action movies so I never watched them.
My dad was a Vietnam combat vet who liked war movies but I don’t ever recall him watching Rambo. Probably too violent for his tastes.
I saw Rocky IV in the theater at the age of 6. I’d watched previous Rocky movies on TV before that.
Michael B. Jordan is a huge star, currently, and was in the Creed movies. I think that turned a new generation on to the Rocky franchise.
I feel like Rocky has more longevity and more mass appeal.
No Rambo movie ever won an Oscar.
That’s true as a movie rocky has more longevity than Rambo. By virtue of the fact the reboot was actually really good whereas the attempts to reboot Rambo over the years sucked.
But as a cultural phenomenon Rambo is far more widespread and long lived. He is the archetype of an action hero (even though he originally was a pitiful figure who’d been broken by war, none of that made it into the archetype)
Thunderlips let him win.
I enjoyed the belated Rambo sequel.
Rocky has a statue
Rambo affected US foreign policy, and led to trumpism.
While I’d concede that Rambo came to define a genre, so did The Blair Witch Project and Dick Dale. That’s not nothing but if you ask for the top “Rambo style” action film, I think that the most popular examples are Commando and Predator, not any of the Rambos. If you look for surf rock, you’ll probably think of the Beach Boys and be, plausibly, completely unaware of the person who invented the genre and (in my mind, arguably, the greater world of rock music in general).
While it may be that Mockumentaries go back to at least Spinal Tap (and probably earlier), I feel like the rise of the format to create shows like The Office, Parks and Rec, the movie What We Do In the Shadows, etc. might follow from the popularity of Blair Witch - the whole style of the endeavor quickly turning to parody.
I don’t know that generating a archetype necessarily makes something the most influential if it doesn’t remain the standard bearer and, particularly, not if the whole genre dies off. I might personally say that Dick Dale changed music more than the Beatles ever did, but I doubt that I could ever raise much support for that position short of paying shills to back me.
In general, I think that Rambo is largely remembered as the caricature of itself that it became and merely one among many of the “shirtless bodybuilder blasting through hundreds of red shirts” genre that’s basically been dead for 30+ years. Once everyone from Gen X and before dies off, the whole concept of a “Rambo” may well completely disappear from the planet, beyond the realm of film historians.
I don’t know that Rocky will go farther but as an Oscar winner, it’s place as a template for building popular, well structured scripts (e.g. Staying Alive, The Karate Kid, Eight Mile, Silver Linings Playbook, etc.), a local icon for Philadelphia, and more success as a reboot, I think it has a good chance to continue into the future both on its own and as the true skeleton for a reliable underdog script. The Rambo series’ basic storylines (after the original), as I recall it, were basically either Old West / reluctant hero tales or Liam Neeson/John Wick style one-man killing bonanzas to rescue or avenge someone.
If someone can say that Rambo forms the basis for John Wick style scripts, I might be willing to revise my viewpoint but I think that goes back to at least 1970s blaxploitation and Eastern (Hong Kong/Japan) film, or Old West films. I think that the Rambo movies are merely an example and - again - not the most popular within the genre.
I believe that Rambo gave us bodybuilder physique-driven over the top action films. Nothing more and nothing less.
Rocky has an edge in the “memorable quotes” department: “Yo, Adrian!” vs. ???
“You have to become war”?
“I didn’t come here to rescue Rambo from you. I came here to rescue you from him.”
“Do we get to win this time?”
“What you choose to call hell, he calls home.”
"Back there I could fly a gunship, I could drive a tank, I was in charge of million dollar equipment, back here I can’t even hold a job parking cars! "
I’m conflicted. I’m inclined to say Rambo, since Rocky doesn’t have his own “Weird Al” video clip:
Of course, Rocky got the parody treatment with The Rye or the Kaiser.
Rambo has decades of imitators, Rocky has a tired franchise. Rocky should be the man all Americans want to be, but it’s clearly Rambo.
I’m going with Rambo.
As a cultural icon, Rocky Balboa represents an individual rising up from difficult beginnings and achieving success. Later movies in the series show the downside of achieving success and losing sight of true values. But the series is ultimately about a person acting as an individual.
John Rambo is a cultural icon on a societal level. (I’ll note that there are effectively two different John Rambo characters, the one in First Blood carries a much different message than the one in the subsequent movies. And I’m focusing on the latter one.) Rambo doesn’t represent struggling against your inner flaws; his problems are always external. He solves problems by shooting his clearly defined enemies, who are always depicting as having started the problem and therefore deserving of their fate. And when he’s shot his way through his list of enemies, the problem is solved. The message of John Rambo is you can solve problems by violence and that complicated problems have simple solutions. This is a message that has caused a lot of problems for a lot of people.
So a person applying Rocky Balboa’s message to their life will be mostly affecting themselves while a person applying John Rambo’s message to their life will have an affect on many other people.
Rocky wasn’t one of Andy Dwyer’s favorite movies.
Um what?
United States Army Vietnam War veteran John Rambo visits the home of his army buddy Delmar Barry and learns that he has died from the effects of Agent Orange exposure. Wandering aimlessly on foot, he arrives at the town of Hope, Washington. Local sheriff Will Teasle spots him and drives him to the other side of town, telling him that he is not welcome
Which of these problems did Rambo solve with the ensuing violence?
Edit: just saw the comment “I’ll note that there are effectively two different John Rambo characters, the one in First Blood carries a much different message than the one in the subsequent movies”. But I’m not sure you can separate that message from the rest.