After a bit of testing, I think I have determined the cause of this bug. It actually has nothing to do with how far away you are from the block. In fact, it seems to have something to do with a bug in Minecraft which has been fixed known as "click mining".
When you mine a row of blocks the "normal" way, standing directly in front of them and holding forward, the system works properly. Say you're mining a row of dirt blocks with a diamond shovel. There is a small amount of delay between each block's destruction and the next block destruction's beginning. If this is hard to understand, basically the arm has to swing back up before the next block can get hit, so there's a small amount of delay.
Someone discovered this and introduced "click mining" to the scene. All you did was release/repress the mouse button after each block was destroyed. This instantly returned the arm to the "unswung" position, eliminating the delay between each block's destruction. Notch figured this out and introduced a small delay after you press the mouse button down, so it negated the time advantage of click mining.
However, when doing the method you described, the block is broken before you're in range of the next block. So really, there's a small period between each dirt block where no block is in range. This instantly returns the shovel to the unswung postion, so there's no delay between blocks. The click mining fix only adds a delay after mouse presses, and this is not a mouse press, you're just holding it down, so there's no delay. Essentially, you've found a way around the click mining fix.
As for tall grass, it still works with grass on top of it. However, you must be aiming for the bottom of the block, because if the tall grass gets in range, the swing delay is reintroduced, leaving you with the delay again.
Long story short, yes, it's a bug. Still, it's sorta useful in select situations, but those situations are rare, so it's not a really big problem.
I suspect you used /gamemode 2 [player] instead of /gamemode 0 [player]. Gamemode 0 will take you back to survival, whereas gamemode 2 places you in Adventure mode, where no blocks can be destroyed, except with the right tools.
Instead of a number for the gamemode, you can use the full name (survival, creative, adventure, spectator) or an abbreviation of the name (s, c, a, sp) to avoid this mistake in the future.
Best Answer
Possible solutions to this question can be found in several places, including this forum post on the Minecraft Forum website. Here are a few things you should try if you are having this problem:
Change Minecraft's performance settings (check here for details on video settings) in Options > Video Settings:
If you haven't already, update Minecraft. According to the forum post in the first link, the reappearance of broken blocks is a known bug for Minecraft versions before 1.8. The forum also suggests that the bug may be centralized around your spawn location, so if you do not wish to update, move about 50 blocks away from spawn.
Overall, checking Java and changing your Minecraft video settings should do the trick. Block reappearance usually means you're lagging and/or your graphics settings are too high for your computer.
Side Note: If you ever have this problem in Multiplayer, the problem may also lie with your internet connection or server plugins.