In the Iron Man animated series, why does teenage Tony Stark have an arc reactor, but then in Iron Man he doesn't get one until a little bit in?
tl;dr: Because the two Tony Starks are different characters.
The Marvel comics universe is huge; so big that it can't even fit into a single universe. There have been so many different versions, revisions, reboots, and adaptations of those characters, that it would be impossible to come up with a single backstory for anyone, let alone someone as common as Tony Stark.
Instead, Marvel's stories are split up into different "universes" within the larger "multiverse". Each universe within the multiverse exists independent of the others, with its own history, characters, sometimes even its own laws of physics. This concept has been explored several times in the comics, most recently with the Secret Wars storyline that kicked off this year, showing all the various universes colliding with each other.
Iron Man, the movie, takes place in the Marvel Cinematic Universe, which includes all of the live-action movies and television shows produced by Marvel/ABC/Disney (that includes all of the Avengers-related movies, plus Guardians of the Galaxy, as well as Agents of SHIELD, Agent Carter, and the Netflix series like Daredevil). Officially, this is Earth-199999. Again, all of those movies and TV shows are internally consistent, but they have nothing to do with the animated series.
The various animated series, on the other hand, generally exist within their own unique, or very limited, universes. Some examples of Tony Stark in various animated series:
- Iron Man: Armored Adventures takes place in it's own unique universe, officially called Earth-904913. This is the only show set in that universe, which is how Tony Stark can be Iron Man as a teenager.
- Iron Man: The Animated Series from 1996 exists within the universe for the Marvel Action Hour (which also includes 1996 The Incredible Hulk animated series, designated Earth-534834.
- A group of Marvel animated shows starting in 2010, including Marvel's Avengers Assemble among others, take place in a universe unofficially referred to as Earth-TRN123 (because it has no official designation yet.)
On a related note, the majority of the comic book issues take place in Earth-616, the "mainstream" universe, while the Ultimate whatever issues take place in Earth-1610, the "Ultimate" universe. That's how, for example, Tony Stark could invent Ultron in the movies, but Hank Pym invented him in the comic books.
Iron Man: Armored Adventures, though one of the more well executed Iron Man series, and inspired by the 2008 movie, does not tie in to that film series or serve as any kind of "supplement" to it. The Real-World reasoning behind the two is simple:
Iron Man the movie was foreseen to be a big hit, and to capitalize on it, execs created a kid's show to extend the familiarity of the brand, and have a possible second line of merchandise to shill.
The answer provided by @kutuluMike which mentions the multiple universe theory holds a lot of weight, .... but its easy for people to get confused as to why a movie franchise would then have a "tie in" cartoon that doesn't follow the original story in some way.
But this is fairly common practice; films tend to have stricter guidelines for audiences and only a certain amount of time to (a) develop character, (b) showcase "must have" items and (c)create memorable action set pieces. Cartoons and live action series, concurrently, tend to have less room for violence and excess, but (a)more time and Space to develop characters, (b)make story lines interesting, (c) get people invested and (d)continue adding new aspects. As such, a lot of movies have "inspired" follow up series of some kind without them being "exact continuations" of the film proper.
Iron Man:AA falls squarely in this category. It was based on the same canon material as the film, and made at least in part because the film proved profitable... but it's not a "Continuation" of the film. Other examples of this happen in both cartoon and live action, usually with a slight change in tone and demographic. Examples of these include:
Batman: The Animated Series directly inspired by Batman 1989 and Batman Returns. The first Batman film was very dark and violent, and while the cartoon doesn't shy away from this, its noticeably less so. However, music, visuals and some story elements (the Joker being named "Jack Napier", for instance, or the look of the Penguin) were all taken from the films.
Buffy the Vampire Slayer the show being a continuation of the film of the same name. The basic story is the same, and this counts more because Joss Whedon, the creator of Buffy, was the driving foce behind both, especially the series. However, this is one instance of the Show being much, much darker and intense than the film, which was meddled with by higher ups despite Whedon's original vision. He had a freer hand in the series dealing with the character he created, so this is why the "Buffy" in the movie and the "Buffy" of the show, though technically the same exact character [with the 5 years between the two being ignored] come across as distinctly diverse when viewed.
Star Wars films inspiring Droids and Ewoks. I don't believe George Lucas would consider the 80s Saturday morning cartoon Droids to be canonical, given where the franchise has gone. And in Ewoks, the little Teddy Bears speak English, and magic is a distinct thing in this series, as opposed to anywhere else in the completely MASSIVE Star Wars canon. It should be noted SW inspired several other pieces of media which had to be incorporated or altered to fit the franchise, from books where Luke has a wife [Mara Jade] to comics, to games which created new characters and Force abilities.
1998's The Mummy inspiring the 2000s cartoon of the same name. In the toon version, Rick and Evy O'Connell look extremely different, being a bit older, and their son found the band of Anubis which first woke Imhotep [finding the band didn't happen til the Second film, and he'd been awakened in the First]. While on supernatural creatures...
1985's Teen Wolf would see a 2 season cartoon on CBS, and even some members of the original cast would play their roles to tie the two together. However, there were distinct differences between the film and the Show; the cartoon seemed to COMPLETELY disregard the occurrences in the film all together! Scott and his dad Harold were still werewolves... but now, they had an extended family of Grandma, Grandpa and a little sister Loopy. They lived in a town called Wolverton, and their lycanthropy was a tightly kept family secret which Scott never made public; only his friends Boof and Styles, whom he'd known since childhood, were in on it. Contrary to being a celebrity, Scott was more of a warmhearted, but social awkward kid, and having to hide his family secret constantly only complicated that fact.
1991's The Addams Family inspiring the 1992 revival H-B cartoon. Obviously, the ABC toon took some cues from the then-popular film (with Gomez and Fester being brothers) and it Completely ignored the birth of the third Addams child, Pubert, from the follow up movie.
2005's Fantastic Four cartoon following up on the 2004 film. There had already been a 1994-1996 series, and both were very similar, sticking pretty close to the comic on which they were based. The similarities to the characters in the latter series shown in the movie, however, were noticeable.
The Toxic Crusaders being loosely adapted from The Toxic Avenger films. The gore and violence from those 1980s Tromo Films B-movies was taken waaaaaaaaaaaaaaaaay down for the short lived Fox Kids 1990s series. Same character and back story, but in the toon, Toxie was green instead of brown, his girlfriend wasn't totally blind, his mop was a semi-sentient living weapon, and he had other hideously deformed mutant allies to help him out; this actually had more in common with the then popular Teenage Mutant Ninja Turtles than its parent series. Speaking of which...
The 1990 Teenage Mutant Ninja Turtles film is an odd mention, because it was inspired by the success of the then-IMMENSELY popular Cartoon, but actually was closer to the Source material of the original comic book! There were just some visual ideas lifted from the toon, such as their differing masks and April's yellow coat, with bits of the cartoon's comedy and lightheartedness. Even when blended with the toon, though, the tone, story and personalities were a bit darker, which put it in line more with the original black and white, creator-pushed comic books.
James Bond Jr. being loosely inspired by the James Bond film franchise. The 007 stories can get intense and violent even before Daniel Craig stepped into the role, with guns, violent deaths and naked women being a norm. The villainous organization SPECTRE were also a terrorist threat not to be taken lightly; in the cartoon, nobody dies, the good guy always wins, there's barely any kissing, and all the super cool spy gadgets are basically non-lethal.And where as SPECTRE was a serious threat to be stopped at any cost, SCUM was more a "bumbling villain of the week" team.
2000s X-Men: Evolution being a "high-school-ified" Saturday morning take inspired by the X-Men film that came out that same year. Many of the same characters are seen, and there's a wonderful amount of action and even depth...but you did not see people dying in the cartoon series the way they did in the movie, and the main characters are younger for kids to identify with. Oddly enough, it could be said the mature themes of race-hate and segregation lightly touched on in the film were actually addressed with more depth in the previous X-Men:TAS from 1992. Yeah, the 90s were a Damn good time for toons.
These are just examples pulled from the top of my head, but many exist. It's rare that you will find a follow up series that captures or closely resembles the aesthetic of its original film, especially if the series is made for kids...but it does happen. One of the best examples would probably be the Highlander series; though the sword play and violence had to be brought down a few levels for censorship reasons, this series ties very closely to its parent film, even going so far as to address issues in the film series as proper canon! And the themes of immortality, morality being gray, comradeship and "there can be only one!" still ring true, even for a TV series.
Taken in that stance, Iron Man: Armored Adventures was never meant to "Follow up" the films or continue them, so these aren't really "inconsistencies", just differences in media that have the same source material, but were meant to play out differently.