I feel like using wallride is never really necessary when playing Lúcio, as just running around normally works just fine. I occasionally do it just for fun running places, but I haven't found any real purpose for using it in game. So, when should I be using wallride, and what advantages does it have?
When to use it? Always!
It is not only for being fancy, and it's not exactly situational either, because it's almost always better to fly around than to walk around like a mere mortal. It's a movement ability, and like all movement abilities it's really good, but this one is pretty hard to learn how to use effectively.
Wall-riding is about so much more than crossing a gap, and that has a lot to do with the fact that you can chain them. That is a bit of a confusing in the beginning of course. You can get an extra jump off the wall and continue on another wall, so in essence, with some practice you can scoot around the top edges of the map once you get a hang of it. Watch some videos like other people suggested, or just practice in a custom game, and you will eventually see that it is very useful for navigating, for escaping or distracting etc.
I can't see anyone that's not using it be a truly effective Lucio, since it's actually one of the best movement skills in the game!
Also, remember that everyone is kind of new to this game. We had quake for a long time before people started bunnyhopping, and this is a thing that requires some finesse, so the wall-riding may well have a lot of potential that we don't know about yet.
'When' you should be using wallride is more of a personal opinion.
To answer your question on its advantages, as well as for the reasons shown in the video posted in another answer, Lucio's wall ride can provide specific advantages over your opponents for a variety of reasons. Planet Overwatch's video shows an introductory guide to Lucio's wall riding ability.
Not only can the ability be used to 'climb' walls in order to get to higher points of the map usually inaccessible to most characters, you can use it glide along walls, gaining momentum and pace as you do so.
Using a combination of some of the techniques show in the video allows you to drop down on enemies from a great height, usually in ways they are not expecting, giving you the element of surprise when capturing zones etc.
There are many guides on youtube showing the basics of Lucio's ability, as well as more advanced video on how to use it effectively in game.
I'm glad you asked the question, because until now I had no idea how to use it effectively either.
One caveat I feel hasn't been mentioned in previous answers is that wallriding increases your field of movement. It's harder to headshot* a Lucio in a corridor than, say, a Tracer if Lucio uses his wallrides, jumps and movement abilities erratically.
I've been able to approach and kill Hanzos and Widowmakers by using wallride jumping while I know they're looking my way.
*unless you're one of those people that can pin a fly to a dartboard while blindfolded and being tickled.
As seen in this video, you can use it to escape the well in Ilios by running around in circles until you reach the top. This can be somewhat helpful if a Roadhog drops you in, but you may end up dying anyway if your team doesn't have control of the point when you get out.
Wallriding by itself is not very usefull.
However when you combine your wallride with well timed jumps, speed boost and Amp It Up you can get to crazy places insanely fast. You can then utilize this to escape certain death situations (as Dragonrage posted), to get to your teammates to help them or to evade incoming enemy fire.