What's the difference between these two types of plugs? Why do they exist? Why can't all plugs be 3.5mm ?
Different form factors allow use for different applications - I believe they are capable of carrying the same signals. 1/4" cables are more durable - I've bent many mini jacks.
The thing to be aware of is that there is professional line level and consumer line level, so you need to make sure your equipment is calibrated to work together.
From your picture, the cable on the left is stereo (trs) and the right is mono (ts) - you can tell by the black bands on the tip. If the 1/4" had the two bands, it could be stereo or a balanced mono cable, depending on how you use it.
See here for some detailed info and history:
I do not know of any meaningful electrical difference between them at the line- or instrument-level voltages that they usually carry.
I believe the 3.5mm plug was created to be a "miniaturized" version of the 1/4" plug, which was already in general use. 3.5mm is more commonly seen in consumer electronics, and it's usually a
I generally see 1/4" plugs used more often in audio production - instrument cables, hardware inputs and outputs, and headphones. I think this is largely due to tradition, but I must say that I generally prefer the larger plug if I'm frequently plugging and removing cables because it feels a little bit more stable to me. 3.5mm plugs require a more precise mechanism because of the smaller size, which strikes me as more delicate.
The Wikipedia link in Sam's answer is also excellent.