"Spontaneously Fermented Ales" and "Wild Ales" are not the same thing. Here's the difference between the two.
The word "spontaneous" indicates that no cultured yeast or microbes were introduced to make the beer. Instead, the beer was made by natural inoculation and fermentation by wild yeast and microbes in the environment (e.g. in the air and wood). That’s how traditional lambic is made.
On the other hand, the word “wild” is often used when a strain of yeast called Brettanomyces is used. Brettanomyces was historically considered a wild yeast, and the word "wild" is still used to describe beer made with lab-grown cultures of Brettanomyces (often with bacteria like Lactobacillus and Pediococcus). Unlike what the word indicates, it’s mostly tamed and controlled in such cases. That’s how most American Wild Ales are made. If done well, it can produce an interesting and replicable beer.
This difference in fermentation methods results in beers with different levels of complexity. Spontaneously fermented ales are like an unplanned party that just happens because the right people are at the right place at the right time. We can certainly create the right environment, but we cannot force the magic to happen. In contrast, many wild ales are more like exclusive invitation-only parties. They likely won’t possess the complexity and nuance of a spontaneously fermented beer.