The difference between the 2 antennas is a 1/4 wave antenna does not produce any gain and is a smaller antenna in height. The 5/8 wave antenna will be taller in height and add gain to the power the radio is putting out.
For VHF(150 Mhz) you usually get about 2db gain from a 5/8 wave antenna. What this means is say you have a 50 watt radio connected to your 1/4 wave antenna, it puts 50 watts out of the antenna(in a perfect world). If you replace the 1/4 antenna and connect the 5/8 wave antenna with the 2db gain rating, it would now be similar to connecting your 1/4 wave antenna to a 100 watt radio.
Since the antenna is longer, there is more antenna to radiate the 50 watts coming out of it, so its similar to having a more powerful radio connected to a smaller antenna, and improves the performance and potentially the range of your radio.
This is the simple explanation without getting into all the tech stuff.
Hope this helps,