What are the advantages of using Leveled Homomorphic Encryption over Somewhat Homomorphic Encryption? Is there a point in using LHE over SHE?

From what I understand, SHE is like an extended version of LHE, the only difference is that you limit the depth of circuits that can be evaluated. Therefore, I wonder which is better to use since they seem to be the same.

What is the effect of limiting the depth of the circuits? That does not make much sense to me.
I would take a look at Brakerski's survey to clarify the exact differences between LFHE and SHE.

In the survey it says


  Leveled FHE [...] refers to a family of FHE schemes that allow, for any depth bound $d$, to generate an instance of the FHE scheme that supports the evaluation of depth-$d$ circuits.


And also 


  In early works on FHE, the term somewhat homomorphic encryption (SHE) was used to indicate a scheme with homomorphic capabilities against a restricted class of functions (depth bounded). The two terms are sometimes used interchangeably, however in the original SHE scheme [Gen09b, Gen09a] the parameters of the scheme grew exponentially with $d$.


So, even though both of them accept a restricted family of circuits, we see that with LFHE you can "plan" in advance which type of functions you want to compute (i.e. up to what depth) and then you can obtain FHE for that family, whereas with SHE this bound on the depth is more 'universal', in the sense that it doesn't scale well with the depth and hence you're forced to keep a relatively low depth (but you can still use bootstrapping if this depth is large enough).

Notice that LFHE actually makes sense in practice: You probably use an FHE scheme with some specific application in mind, and in that case you can find a suitable LFHE scheme that fits your function.