Minecraft – the difference between powering a block with a repeater and with redstone dust leading into it

Redstone Lamps (unlike glowstone and pistons) act like opaque blocks, not transparent ones. Opaque blocks have two main differences from transparent ones:

1. They power adjacent opaque blocks when given "normal" power (like a redstone wire running into the block), and they power adjacent transparent blocks (including redstone wire and pistons) when given "hard" power (like a repeater running into the block or a torch underneath the block). They will not power any adjacent blocks when given "soft" power (like power from an adjacent redstone torch).
2. Redstone wire, redstone torches, and other objects that "attach" to blocks can be placed on redstone lamps.

This means that redstone lamps behave mostly like any other solid block in Minecraft when it comes to how they are powered, so you can power them many different ways:

That configuration on the left is powered by a redstone torch under the middle block, like so:

While having redstone lamps be opaque opens a great deal of possibilities, it also has the unfortunate downside of making it impossible (unless there's a method I'm not aware of) to turn on a single redstone lamp in a floor made out of lamps from below, so keep that in mind when building structures with them.

The confusion lies in the fact that there are two different types of power a block can emit. These are sometimes referred to as strong and weak power.

Weak power is given to a block under the conditions mentioned in the question, by having wire on top of the block or pointing into it. This will unpower torches, extend pistons, play note blocks and turn on lamps & repeaters.

Strong power is given to a block by having a lever on it, a torch underneath it or a repeater pointing into it. In addition to the abilities weak power has, it will also power redstone dust in an adjacent block.