These are different games
Okay, it seems nobody has said this one yet, so, but one thing that should be clear to you: 4e and 5e are quite different games. Actually, 5e is probably closer to older editions than to 4e.
As for being a new player in 5e, this is easier than it used to be. 5e is quite newbie-friendly. In particular, I feel like the other answers fail to address the fact that you (at least from what you said) are going to play 5e, not 4e.
TTRPG x MMORPGs
Same thing goes for WoW vs D&D: They are massively different. As already explained to you, a healer in D&D is more often buffing and supporting than actually healing - especially (but not only) in 5e, preventing damage is better than healing damage. All the aggro in WoW goes for the tanks - both the aggro system and the tank skills are designed for that. One melee hit from Gul'dan (sorry, most recent example for me, stopped playing before Argus ;P) and the Mage is popping Ice Block or dead. In D&D, even a squishy wizard can actually take a few shots (except at level 1-2, they have about less than 10 HP there, that's an easy goblin kill lol). Point being: different games, a few common concepts, but different games (admittedly, 4e is a lot closer to WoW than most other editions, though). Another important distinction is that while a Fury Warrior will always be a DPS, the same class and subclass in D&D can be a lot more diverse, depending on how you build them. The same Assassin Rogue can be ranged or melee, the same Champion Fighter can be an archer, a tank or a duelist. Find whatever you feel more comfortable playing, don't be scared of trying stuff out.
Now, about your fears, there are a few things that you can do.
Read some guides
You have read Icyvein, WoWhead or Noxxic before so you could figure out your rotation, what legendaries are top tier for your class and maybe guides for boss fights. Maybe now you are a top tier player that can figure out your own rotation, are in a mythic progression with a guild that figures out the boss fight strategies by themselves and can do your own math on how much DPS your legendaries will give you, but probably at least once before you read these guides. It's not that different for D&D.
You most likely won't find guides here on SE, but check out our list of forums - there are lots of good guides around. One I will always recommend is the Treantmonk's guide for God Wizards, even if you don't intend to play a Wizard, it gives a nice summary of party roles, both in combat and out of combat. Read guides that actually explain their insights and the role of that specific build - guides saying "Take this one because I like it" or simply "This is better, period." might help you with the build, but not with the concepts involved.
Ask for help
You've already started doing so here, but it will be easier, faster and most likely more efficient if you ask for help from your DM and your fellow players (especially if they are more experienced). Tell them exactly what you would tell us - "I feel like I am not doing my job right, can you help me? What am I doing wrong? What's happening here?"
If you feel like someone else is screwing up, talk to them - kindly. It might be that you guys are not on the same page, but both had good intentions.
If you need to, play it easier
Although Spellcasting has been greatly simplified, spellcasters are still harder than usual martial fighters. You might want to start with classes with little or no spellcasting at all - Barbarian, Fighter (not EK), Rogue (not AT) are quite straight-forward to play. Monks also aren't that hard but already have a lot more options for your actions, which is interesting and harder at the same time. Spontaneous casters like Sorcerer and Bard are usually easier to handle than prepared casters like Clerics and Wizards, too. If you are having too many difficulties with the mechanics, don't add more problems with the character mechanics as well, learn the things one by one. I'm not citing WoW examples here because people get mad every time we say a class is easier than another heh.
Either way, obviously don't let that spoil your fun - playing something "badly" while having fun is always better than playing something like a boss and waiting for the session to be over because you are bored as hell.
Most important thing to remember: The only way to play TTRPGs wrong is if you are not having fun.
This is the core of TTRPGs. D&D5e has another core, which are the only things you should be too worried about:
- The DM describes the environment.
- The players describe their actions.
- The DM describes the results of their actions.
That's all. Except if you are playing in organized play, the rules are for helping you with that, not to force you into something. Have fun, be sure you are not screwing anyone else's fun, keep having fun, and that's all. Anyway, the point here is: don't worry too much about screwing up - sure, you weren't having fun feeling useless, but that feeling can be changed.
And my final statement on that matter, that I've used before and got lots of arguments: You don't need to be useful (and you can't be useful every time, at least not if the game is balanced enough), you need to have fun with your group.
Lack of Synergy
In one of your comments, you mention your actual main concern is the lack of synergy between characters. This is also an easier thing to do in 5e, IMO, as well as not that much of a problem. Unless someone is constantly and intentionally making things that preclude your own character, you won't have this problem, at least I never had nor my players. Either way, the most important thing here is talk to them while making your characters. When I DM (which is most of the tables I play, sadly for me :P) I ask my players to build their characters together and with me, so we essentially have a Session 0 that includes character building. This lets you directly talk to each other "Hey, my character has this concept, it works like this, it would help if you could do that". If, during the game, something does not work out as you planned, first: is it actually a problem or just an opportunity to make things funnier? Second, if it is a problem, just talk to them "So, bro, I assume you are having fun doing X, but that's stopping me from doing Y, how can we work this out?"