While neither the tv series or the comics discussed age, they presumably have a different age in each media. In Teen Titans(TV series), Robin is theorized to be 16. In The New Teen Titans(the comics), he goes to university, putting his age roughly at 18 and older. While the different styles between comics and animation have to do with the character appearances and whether or not they look their age, I think that the creators of the tv series purposely made them younger to relate more to the target audience. In the most recent adaptation(a live-action version), they're bigger age gaps of the Titans, and Robin is 30.
There are a couple of differences among the two Robin. Starting with the most significant difference, Comic Robin is wearing his signature green leotard while TV Robin wears black pants. The design change isn't surprising since Robin, as a character, went through a couple of costume changes, but TV Robin resembles Tim Drake's costume created by Neal Adams in 1989 more than Comic Robin. Another difference is the color of their cape; Comic Robin's cape is yellow, and TV Robin's cape is black. Lastly, Comic Robin wears green boots while TV Robin wears grey boots.
At the beginning of both series, they have similar personalities: good leader, stoic, stubborn, and usually serious with a few funny moments. However, they go through different developments. Comic Robin becomes detached from the other members, usually lashing at them when he's under stress. He started distancing himself from them, refusing to share what was causing him stress with the other members, constantly insisting that he didn't need their help, etc. until it caused a strain on his relationships with the other Titans. The strain in his relations with the Titans leads to his inevitable leave from the group and becoming Nightwing. Now, TV Robin never actually becomes Nightwing in the series(there being a slight glimpse of Nightwing in the dystopian future Starfire visits), so his development isn't building to an inevitable leave. His development is positive, going from self-reliant to being able to rely on the others. He gets closer to the other members and becomes involved in their problems. While he doesn't have any issues in the tv series, he does open up unlike his comic counterpart.
To start explaining his character's background, both TV Robin and Comic Robin is Richard Grayson. They're multiple Robins throughout the DC history, but for both media, they stuck to Dick Grayson. Robin's backstory is never really explained in detail in either the tv series or the comics. Through research, we found that they were the same in both. Robin was born to John and Mary Grayson, and they all lived as trapeze artists at a circus. A mob boss called Tony Zucco, who threatened the circus, ended up calling a hit on Robin's parents when the circus' owner refused to pay him. Batman takes him and trains him to kill Zucco; he ends up staying as his sidekick. After a while of working with Batman, he leaves and ends up in the Teen Titans. While they're some adaptations where Robin created the Teen Titans, Raven is the one who gets them together in the comics, and in the tv series, Beast Boy is the one who suggests that they become a group. In the comics, he's in college and lives with Batman even though he's in Teen Titans for a short time before moving out, presumably moving into the Tower with the other Titans. He's also aware of the existence of another Robin; the next Robin was Jason Todd. In the tv series, Robin has already left Batman and has no association with him; there's no mention of his time with Batman.
Cyborg: They aren't close in the comics; their only interactions are when other members are present, or Robin is asking for a status report for like the T-Car or the T-Plane. They don't have a reason to interact outside of the group, and their relationship was never really developed. However, in the tv series, they're pretty close. Robin tends to confine in Cyborg when it relates to plans dealing with the Titans or what next move they should make. They like working out together, and since they share an interest in machinery, they tend to bond over that. They even have a special move together that they use to battle together. I think another reason they're close is that Robin can share serious moments with him. I would argue that he's closest to Starfire, but Cyborg is who he's closest with next.
Raven: Similarly to Cyborg, Robin didn't have much of a relationship with Raven in the comics. He was more understanding of her whenever she was being secretive and offered her a shoulder whenever she seemed distressed. In the tv series, Robin and Raven aren't that close until the fifth season, Raven-centered season. Robin shares an understanding of feeling powerless and constantly reminds her how important she is to the Titans. Even when she tries to hide from the others, he still looks for her and goes to great lengths to help her. Raven and Robin both like peace and can share a room without having to feel it with noise. It also tends to be able to read Raven better than the other members, an example being during season 2, episode 5, where he figured out Raven was scared even though she said otherwise. She even hugs him which I think is her way of showing that she trusts them meaning she trusts Robin.
Beast Boy: In the comics, he has high expectations of Beast Boy and gets frustrated when Beast Boy isn't serious. Beast Boy looks up to Robin and is impressed by him, but because of the pedestal he puts Robin on, he feels inadequate. They don't sync up all the time, so they misunderstand each other a lot. He takes the role of teaching Beast Boy because he's older, and while Robin's not verbal about it a lot, he thinks well of Beast Boy. Even considering all of that, Robin doesn't have that much of a relationship with Beast Boy. In the tv series, I think they're the least close. Their personalities are pretty different, so they tend to spend time with other Titans before they seek each other out. Robin's focus usually isn't on Beast Boy even when something happens to Beast Boy.
Starfire: There's a reason I left Starfire last, and that's because there's a lot to unpack here. In both the comics and the tv series, they're romantically involved; however, the way the storylines proceed are distinct. They get together fairly quickly in the comics. If I'm honest, I barely remember any development in their feelings to make it believable that they wanted to be together. For a while, it's going fairly well, but then they go through some issues. During his development of distancing himself, he's also distancing himself from his girlfriend, which leads to large strain on their relationship. Starfire, wanting for the relationship to work, constantly follows what Robin says to avoid a fight. Even when he becomes rude towards her, she stays on his side, trying to portray that she wants to help without seeming pushy. There's not much else on the relationship in the comics, but I looked up a history of them: Click Here for More. Sadly, they didn't have a happy ending, unlike the tv series. I want to point out one similarity they have, and that's when Starfire has to go home to marry someone. In the comics, she marries him, but in the tv series, the Titans, including Robin, stop the wedding. In the tv series, the storyline is pining best friends. For five seasons, they dance around their feelings, and then they get together in the movie. He's very close to Starfire and is always worried when something happens to her. He seems fond of her and is always first in line to comfort. He also helped a lot in teaching her the way of humans. The other Titans tease him a lot about his behavior around Starfire.
Terra: TBD
Wonder Girl/Kid Flash: Wonder Girl only shows up in the comics, so I'm going to be talking about their relationship only for that media. Wonder Girl is one of the few members that Robin knew personally before the creation of Teen Titans; they even sometimes fought together in battles. She tells Robin about her problems and worries the most besides Starfire when he becomes distant. She's not afraid to call him out on his rude behavior, and he trusts her the most to lead the group if something happens to him. While they get into fights, they care a lot about each other. Now, Robin and Kid Flash didn't like each other in the beginning; similar to Wonder Girl, Robin knew him personally before Teen Titans. However, they tended to fight each other since they had conflicting opinions a lot. Kid Flash doesn't do well with authority, and he saw Robin as an authority figure against him. As the series progressed, I think they ended up having a better understanding of each other because they were sharing the same struggle: balancing school/work alongside the work of Teen Titans. They both end up leaving Teen Titans. In the tv series, Kid Flash is an honorary Titan and doesn't interact with the Teen Titans. He had his subplot that includes Jinx, who later becomes his girlfriend and doesn't stop being a villain.