What's the difference between a Forbidden card and an Illegal Card?
I understand that the difference is that you can use Forbidden cards in a private duel if both players agreed to it before, or in a Championship if its regulations allow it. But you can never use illegal cards. Why not?
There seems to be a bit of misunderstanding between what an "Illegal" card is and what a "Forbidden" card is.
Basically, a Forbidden card is one that was at one time legal to play in OCG or TCG Advanced Format tournaments, but are no longer legal in one or both of them. Note that the TCG also has the Traditional Format, which does not have a Forbidden List.
An Illegal card is one that was never allowed to be played in a tournament. All of the cards have the text "This card cannot be used in a Duel" or a similar phrase written in the bottom left corner, aside from four tournament prize cards that are also considered illegal. Also, counterfeit and unofficial cards are automatically considered illegal. Illegal cards are the only cards that are not allowed to be played in the TCG Traditional Format.
Why are cards made Illegal?
While you can't use Illegal cards in a tournament, there is nothing stopping you from using them in a private duel with your friends. But almost all of the Konami-produced Illegal cards are such because they are pretty much unplayable (by which I mean it is impossible to have a meaningful, fun duel by playing them).
Quite a few Illegal cards, such as Legendary Dragon of White are Match Winner cards meaning that you automatically win not just the duel, but the entire match if you bring your opponent's life points to zero with a direct attack from that monster (tournaments are usually held with three duel matches). The problem was that players would surrender before the player with this monster resolved the attack, which led to accusations of bad sportsmanship. So these cards were banned from tournaments, and Match Winner cards are useless in a casual game with your friends since you aren't playing in a tournament setting.
Other ones include ones that are only playable if you attended a certain tournament (e.g. Get Your Game On!). I imagine that these are Illegal because someone could demand you show proof you were there, which pretty much stops a duel in its tracks.
There's also cards like Card of Last Will that were based on absurdly powerful cards in the anime. Basically, there wasn't really a way to make it playable and still resemble the anime effect, so they just made them as Illegal cards to be fun promo cards (and it says a lot that this card is too powerful given that it's actually weaker than the one the anime had).
So making a card Illegal generally is Konami's way of indicating that "this card is unplayable even in casual duels with your friends".
Why are the Egyptian God Cards Illegal?
Regarding the Egyptian God Cards, the originals were also unplayable. For instance, The Winged Dragon of Ra had
ATK ? / DEF ?, but the card didn't tell you what that meant or how it could change. Same with Slifer the Sky Dragon, which had
ATK X000 / DEF X000 with no description of what that meant. Obelisk the Tormenter at least had
ATK 4000 / DEF 4000, but no other abilities to make it useful, and the blue color might have caused confusion with Ritual Monsters. While it might have worked as a high point Normal Monster, I suppose they probably just made it illegal to match the other two Egyptian God Cards.
Put another way, these Egyptian God Cards are just fancy pieces of promo art that don't have enough rules on them to be playable.
But I heard that there were some legal Egyptian God Cards!
While the original prints were Illegal, Konami made rereleases of these Egyptian God cards that are actually playable and are completely legal to play in tournaments (and they lack the "This card cannot be used in a Duel" text). They are all Effect Monsters and have their abilities explained (which are more balanced than the ones in the anime). Check out The Winged Dragon of Ra, Slifer the Sky Dragon, and Obelisk the Tormenter. Most of these were distributed as promos: I got the legal version of Obelisk when I went to see Yu-Gi-Oh! The Dark Side of Dimensions at my local movie theater.
Will the Egyptian God Cards destroy the world if I play them?
In the anime they said that the Egyptian God Cards are so powerful that if someone loses the control over them in a duel, then they can destroy the world.
Is that the official reason?
I can assure you, Konami does not make the original set of Egyptian God cards Illegal because they fear that a random duelist will summon the card and lose control of them, inadvertently destroying the world. I have played my Obelisk before and the world is still intact. But then again, maybe it's only a risk if I grow hair as crazy as Yugi's ;-)