An ordinary 1/4 wavelength vertical is smaller and resonant without any loading coil or matching network. What's the advantage to a 5/8 wavelength vertical? Why 5/8 in particular, and not something longer or shorter?
Indeed, why? A 5/8λ isn't resonant where a 1/4λ or 1/2λ would be.
The reason is the radiation pattern. The pattern for a 1/4λ monopole is essentially a doughnut, tasty and a pretty good pattern especially for a VHF antenna. Extending the antenna changes the current distribution. This flattens out the pattern removing power from the useless (for VHF purposes) vertical dimension and giving more horizontal gain and at a lower angle. See the following illustration from the late great L. Cebik:
Depending on the source, they will quote anywhere from 1.2dB to 3.5dB gain over the 1/2λ design. There has also been some discussion that in some areas (urban and mountainous terrain) the lower angle of radiation is a detriment and a standard 1/4λ or 1/2λ antenna is to be favored. I don't know. I guess they are pretty cheap, if this is a concern you can buy one mount and two and antenna, compare them for a bit and return the loser.
So, why 5/8λ? Why not long longer? After all more gain is better right? Well, inspecting the figure above you will notice the appearance of high angle lobes. As you lengthen the antenna past 5/8λ these lobes become more pronounced and break up the pattern in undesirable ways. Making it shorter maintains a good pattern, but the gain is less. So, 5/8λ is about optimal for this style of antenna.
A 5/8 wave provides a style of vertical radiation pattern which doesn't offer energy in some of the mid- upper angles but compresses that 1/4 wave donut like stepping on it, to being flatter but WIDER.
If you're attempting to contact a station 10 miles past the donut of energy created by the 1/4 wave, then you won't be heard, if you then switch to a 5/8 wave you will concentrate the transmitted power into a flatter-lower angle and wider donut pattern of energy and now you might be heard where you weren't making it before when using the 1/4 wave.
The downside is the lack of midrange energy at the medium upper angles and is where a 1/4 wave might be the better choice when attempting to transmit over a range of nearby hills which could block and deflect the low angle energy of a 5/8 but the upper angle energy of a 1/4 wave might propagate successfully over that range of hills or obstructions.