What are the effects of discovering a natural wonder?

Natural wonders don't directly provide a boost to amenities/happiness as they did in Civ5. As Auditore notes the immediate effect is an experience boost to the scout.

Natural wonders are also important if you are playing for Religous Victory because there is an Apostle promotion called Pilgrim that gives 3 extra spreads once the Apostle stands next to the natural wonder for the first time. It can be hard to find them on the map when you need to do this, but the trick is to use Strategic View, where the Natural Wonder will look exactly as in the icons Auditore posted!

Natural Wonders also come into play for at least one Great Person. Charles Darwin (Great Scientist) has a bonus that grants 500 science for each adjacent natural wonder. There are some natural wonders that extend over many tiles so you can multiply that by 6 or 7.

Natural wonders are also one of the possible locations for a National Park (which is unlocked by Conservation and provides extra tourism and amenities). National Parks must be designated on squares which are mountains, natural wonders, or have an appeal of charming or better. I don't think you have to be able to move a unit onto the square to do this given that mountains also count, so I don't think you need to worry if you can't walk on top of the natural wonder (need to confirm).

In summary, Natural Wonders don't play one specific role in the game, but they come into play for lots of specific situations. I'm sure there are many situations beyond just the three I mentioned here.

I am not sure if they also increase appeal for neighborhoods or not, but in any case that factor would already be set.

Very good summary, there is also the fact that sometimes it comes a city state mission (to discover a natural wonder to get an envoy) There is also a random bonus xp to apostles - it basically says that when you lead him beside a natural wonder it gives +3 spreads I reckon