Arguably the biggest difference between the "classic" and "modern" examples is the animation medium. Cowboy Bebop was created using painted cels, and Fullmetal Alchemist is digital. A better distinction is "traditional (or cel)" and "digital." The art style can also be tacked up to different designers and, again, the medium.
Evangelion's not
really a fair comparison, seeing how it went from cash-strapped TV series to high budget movies. Even newer (digital) TV series face budget issues and poor animation. Movies, naturally, are shorter and have much higher budgets. There are also OVAs, which are generally shorter and can have better animation than the average TV series (though rarely as good as movies). There aren't really any shot-for-shot remakes out there as far as I know. Rebuild 1.0 and I think 2.0 have some scenes reanimated, the Macross opening was reanimated in 2012 (and looks pretty awful, in my opinion) and I think there are some Gundam shots that are either really similar or are redone? Those are all I'm really aware of, though.
Generally, when I hear "classic," I think something like...AKIRA, Ghost in the Shell, Lupin III, Mobile Suit Gundam. And Cowboy Bebop, of course. I don't think it's really an indicator of age or even popularity, more like something that has stood the test of time. There are a lot of (as I've heard it aptly put) "flavor of the month" series that gain large followings in a short time but are really not that great and fizzle out when the show's not actively airing. There are also a lot of shows that get largely forgotten, through age or lack of licensing, but ARE good. Of course, I can really only speak of what I've observed in the Western market.
As for my personal recommendations?
Giant Robo is quite accessible to casual or new viewers and, for what it's worth, it's been dubbed (twice!). Its main, glaring issue is its incompleteness. Thanks to a lack of budget, it only has seven double-length episodes and covers the penultimate arc out of what was supposed to be a much longer series. Still, it deserves all the love it gets, and I know a number of forum members can back me up on this. In a similar vein, perhaps you might like
Big O. It's basically what would happen if Giant Robo and Batman: TAS had a sexy lovechild.
Memories (Otomo Katsuhiro's Memories) is a set of three forty minute films. They're not connected at all and differ quite a bit in style and theme, so you can watch one at a time or consecutively.
I'd also highly, highly recommend
LoGH. Between its length and its complexity, it can be rather hard for some people to get into (which didn't stop me from charging headfirst into it when I'd never so much as finished a series before, so it really shaped my tastes a lot). Still, it's
the series I'd recommend to everyone.
It all depends on what you can handle. If you're not one to get squicked out by violence or fanservice, I can recommend others/second a number of titles already listed.