Why doesn't diffusion take place between immiscible liquids?
In gaseous phases, we generally assume that there's very little or no bonding between individual molecules (or atoms in, say, a noble gas). This isn't strictly the case, but it's a pretty good approximation, and many good approximate techniques rely on it. A gas that behaves entirely in this way is known as an 'ideal gas'. (See also: ideal gas law.)

In contrast, liquids have bonding between individual molecules, and their behaviour is strongly affected by molecular properties.

So, in the examples you gave:

1. Bromine and air

In this example, imagine that you have a bunch of ping pong balls bouncing around at a reasonably high speed. (Ping pong balls are often used as examples for ideal gases.) Since the balls are already travelling at high speed (given a sensible temperature), and they only get in each other's way by bouncing off each other, it's relatively easy to get the bromine and the air 'balls' to mix, and there's really nothing apart from gravity to make them separate again. 

2. Water and oil

Water molecules form pretty good bonds to other water molecules. (See: hydrogen bond.) Bonding in oils is generally weaker, but also occurs via a very different mechanism to the bonds in water. (See: Van der Waals force.) This means that oil does not form good bonds to water, and any water to diffuse into the bulk oil would have to break the bonds to all the neighbouring water molecules, and any oil to diffuse into the bulk water would have to separate a few water molecules to make a space to do so. Not just that, but since the water and oil molecules can reform their bonds if they manage to connect to other water and oil molecules, if you mix them up, the droplets will tend to recombine into the separate phases.

As a comparison to this example, let's check out two liquids that do mix.

3. Water and ethanol

For the water, we have basically the same situation as before -- water molecules forming good bonds to each other. The ethanol, though, has an -OH group that can form bonds to the water in the same way that the water does (though not as well). This means that ethanol that mixes with water (and vice versa) will tend to stay mixed, and given that the liquids are being mixed around just by random motions, means that you'll get one mixing with the other just as a matter of statistics.