Will I see a noticeable difference switching from a .4mm to a .3mm nozzle?
1) Smaller nozzle advantage: sharper "corners" (higher X and Y resolution)

2) Larger nozzle advantage: faster 3d printing (because you can print the shell faster as each perimeter can be thicker so you'll need less perimeters to be printed to get the same shell thickness. Same true for infill).

3) Smaller nozzle disadvantage (varies, debatable): higher risk of clogging. There are of course other factors which can result in clogging, but this can also affect it as even finer particles in less quality filament can get stuck now.

4) Smallel nozzle disadvantage: since less plastic can come out at a given duration of time, it means less speedy printing of larger layer heights, if at all possible.

I feel like I should get back to (1) and explain why it is so.
When companies market their 3d printer they usually talk about the layer height (the Z resolution) completely ignoring the X and Y. This makes sense for marketing. Luckily it is not hard to explain what is usually left out and what is important for one of the answers to your question on what are advantages of smaller nozzles:

The Z resolution depends on two main things:
1) The stepper motor driver accuracy of your 3d printer's motherboard. Pretty much all are the same now. Claiming your 3d printer can print at 20 micron layer height doesn't say anything, pretty much any 3d printed today can.
2) Nozzle/extruder quality, nozzle diameter. The latter surprisingly does not determine much. Because of how FFF/FMD 3d printing works, the perimeters of your 3d print can be way thinner than your nozzle diameter. This might sound odd but there's a simple explanation: the molten plastic coming out of the hotend is squished and stretched when the head moves. Because it is stretched, it can be made thinner as you increase how much it is squished. There's not much experimentation done on this and available online, only from personal experience I can say I can print perimeters 50% thinner than my 3mm nozzle. So I suppose if you wanted thinner perimeters than that, you would need to swap to a thinner nozzle.
Why would you want thinner perimeters? To get sharper corners or in other words better X and Y resolution for your 3d prints which is ignored by most unlike the overstated "layer height" (Z resolution).
Here's an illustration to help you visualize how fatter perimeters and less sharp corners/edges on your 3d model result in "lower X/Y resolution" (the illustration is a 2d cutout, viewed from the top):