Yep, they should have basically the same content, as far as I'm aware. It's just that the one with the simpler cover (this one, I mean) is a recent reprint of the 3.5 PHB.
Anyway. Don't forget that you'll also want a Dungeon Master's Guide as well as a Player's Handbook and Monster Manual. (Or, if you're not planning to be the Dungeon Master yourself, you can skip the MM and just get the PHB.)
2E books are treasures--you are smart to hold on to them.
For the 3E and 4E stuff you are planning on getting rid of, best thing is to ask if there is a kid in your extended family who might want the collection.
You could check to see if amazon offers trade-in value for them. Like here https://www.amazon.com/Dungeons-Dragons-3-5-Players-Handbook/dp/0786962461/ref=sr_1_1?s=books&ie=UTF8&qid=1483034452&sr=1-1&keywords=Player%27s+handbook+3.5
You see they are offering $13 on an Amazon gift card. It is easy, you just put them in a box and send them in in one large trade-in batch. The book will find its way to someone who needs it eventually, because it still has value.
Many don't offer trade-in, like the 3.0 core rules, but that's because the used market is selling them for a couple of bucks. Nothing to do about that, WOTC's plan was to obsolete some books to resell core line, and they did a good job. There are simply too many in the market for the demand, and it isn't going to pick up for those editions. Probably best to just throw them out--it's sad, but you will probably spend more of your time and gas trying to get them somewhere than they will ever be worth. Content yourself with knowing that there are many, many more copies of them out there than will ever be used.
I personally would go wih Amazon trade-in over eBay selling because it is hassle free, but it all depends on how much you value your time. I would also figure most libraries would determine that since there is a newer edition out, and this is outside their core line anyway, they wouldn't be likely to keep it. Amazon trade-in is just a hassle free way of getting it out into the marketplace, increasing supply for those who want them.
But certainly keep your 2E stuff, unless passing it on to someone you know will love it. It connects and is compatible with 20 years of AD&D material across 10 campaign settings, and is the last of the Gygax style old-school RPGs, so it is something that can never be replaced by the new stuff.
if you're going to play 3.5, you'll need this, if you're going to play pathfinder, you'll need this. You could also refer to the D&D 3.5 SRD or the Pathfinder SRD, but personally, I prefer having an actual book in my hands. :)
Edit because formating fail ...
>I know how hard it can be to get your hands on these
Not that hard. amazon.com is selling them for under $33 each (retail $50), and they qualify for Prime shipping: