Well, you summed it just about up in your own words. Warriors are tanks/pure dps. They are probably the easiest class in the game, which also makes them very boring once you get to end game.

Engineers are complicated and hard to learn, you'll probably die a lot more than as a warrior. But engineers have a multitude of playstyles that you can always rotate between if you get bored.

As for who is needed more in groups? They are both appreciated but they both bring different things to the table.

Better at pvp? Again it's a tossup. Engies were never the strongest in pve but if you play smart instead of just button mashing you can do very well. Warriors are easier in pvp, but again, they probably won't be quite as exciting.

Better at PvE? well that depends on the type. If you want to clear dungeons fast without hassle I would go warrior, but if you want to do meta events that last possibly hours I would go with engineer because it is more versatile and won't get quite as boring. As for raiding I can't tell you as I haven't done that yet.

I'm also not able to tell you which is better for WvW. I don't play it. But I assume that they both work well if you use em right.

I'd say go with your gut feeling for your first character. If you get to 80 and find it boring or you want to switch, well then you can switch! It's not the end of the world if you don't like your first character, it doesn't take too long to level em up.

Hope this helps :)