One more line where 4...d6 is better:

In the Four Pawns Attack White sometimes tries the sideline 1.d4 Nf6 2.c4 g6 3.Nc3 Bg7 4.e4 d6 5.f4 0-0 6.Nf3 c5 7.dxc5!?, which has some venom. But when Black has played 4...d6 instead of 4...0-0, he can avoid that line by going 5.f4 c5(!) 6.dxc5?! (6.d5 0-0 should lead to the 4PA main lines) 6...Qa5 7.Bd3 Nfd7!, when White is already struggling to equalize after either 8.Bd2 Nxc5 or 8.Nge2 Nxc5. Credit where credit is due: I first saw this point mentioned in a chapter by Glenn Flear in Dangerous Weapons: The King's Indian.

Looking at the database now I see people are using this line as a drawing weapon: 5.f4 c5 6.dxc5 Qa5 7.Bd2 Qxc5 8.b4!? (else White is getting an inferior version with the Bishop on d2; in the normal 7.dxc5 variation his standard setup is Nf3, Bd3, Qe2 and Be3) 8...Qxb4 9.Na4 Qa3 10.Bc1 Qb4+, etc. Though Black could (and maybe should) keep the game going with 8.b4 Qb6 instead.

Actually the already mentioned book Dangerous Weapons: The King's Indian by Palliser, Flear and Dembo (Everyman, 2009) is uniquely relevant to this discussion - it also has two full chapters based on a delayed 0-0, which obviously require 4...d6:

* The Hungarian/Kramer variation with 1 d4 Nf6 2 c4 g6 3 Nc3 Bg7 4 e4 d6 5 Nge2 Nbd7 6 Ng3 e5 7 d5 h5!?, with ...Bh6 often on the agenda; 

* The Seirawan system with 1 d4 Nf6 2 c4 g6 3 Nc3 Bg7 4 e4 d6 5 Bd3 Nc6 6 Nge2 Nd7!? using the tempo "saved" on 0-0 for a faster-than-usual ...e5 and ...Nd4. 

Both of those were by Richard Palliser.

But there's also a chapter where ...d6 is delayed: On 1 d4 Nf6 2 c4 g6 3 Nc3 Bg7 4 e4 0-0 5 f3 Nc6!?, by Yelena Dembo.

In practice I've also seen a delayed ...d6 used with the idea of sometimes going ...c6 and ...d7-d5 instead. This has to be dubious against most setups, but maybe it makes some sense against the Averbakh, where there is a known Black setup with ...Na6, ...c6 and ...d6-d5.
I actually faced this 4...0-0 with ...c6 and ...d7-d5 ploy once when I surprised a strong opponent with the Four Pawns Attack, but I easily got a stable space advantage and attacking changes.

Finally, 4...0-0 5.Nf3 c5!? may be a valid move order if Black is after a Maroczy Bind or a "Lazy Benoni" with the delayed ...exd5. The question is whether Black should be after those things...
Instead 6.dxc5 Na6 seems to work out OK for Black after 7.e5 Ng4 8.Bf4 f6 or 7.h3 Nxc5 8.e5 Ne8 9.Be3 b6. (Warning: this is very cursory analysis.)

Thanks for the info, an ordinary chessplayer!

The only difference I'm aware of, for the _white_ side, is 5.Bg5. Here is what Hartston wrote about the "Accelerated Averbakh" 
Quote:

Looking at "note (c)", it's not one line. Hartston says "White has a number of promising plans":

• (c1) 6.f4 Nikolayevsky - Vasiukov, Vilnyus 1957 (-/+)
• (c2) 6.Bd3 Gligoric - Stein, Lvov 1962 (+/-)
• (c3) 6.Qd2 Szabo - Fischer, Leipzig 1960 (=)

Of course Hartston was writing when 5.f3 O-O 6.Be3 c5 (!) was unknown, whereas today it's a main line. In other lines Qd1-d2 wins an important tempo by attacking a pawn on h6, but in the ...c7-c5 lines black defends this tactically, or even advances ...h6-h5, transposing to a position usually reached by ...h7-h5 which is a normal move in the structure. Nowadays I don't think black should be so afraid of the Sämisch transposition, meaning it's an ordinary Sämisch rather than one that is particularly good for white.
4...O-O 5.Bg5 h6 6.Be3 d6 7.f3 c5:

• 8.dxc5 The addition of ...h7-h6 is unlikely to harm black, although I'm not an expert and perhaps ...Bg7-h6 is a missing resource in some offbeat variation.
• 8.d5 This is a direct transposition to the 6.Bg5 Sämisch theory, where ...h7-h6 Bg5-e3 is routine for both players.
• 8.Nge2 In this line I think black might actually be _helped_ by having played ...h7-h6. There is one variation where white plays Be3-g5, but that's not on the cards now.