since from what i understand SHE is like an extended version of LHE
Your understanding is, unfortunately, not correct.
What is the effect of limiting the depth of the circuits? it does not seems to make much sense to me.
I am not sure if you got the point, but the limitation imposed by SHE schemes was not intentional, the authors didn't propose homomorphic schemes with such limitations because they though it was better like that, it just happened to be the only way people knew how to construct homomorphic encryption schemes at the time.
So, just highlight some differences between leveled and somewhat homomorphic schemes:
Squashing the decryption circuit
Somewhat homomorphic encryption schemes are schemes that can perform two operations homomorphically (basically, addition and multiplication), but only a limited number of them, but the important point is that you can't increase this limit by choosing new parameters. That is, if a scheme is somewhat homomorphic, it has an intrinsic limit in the depth of the circuits it can evaluate, typically, $O(\log \lambda)$.
It is because of that limitation that the first SHE schemes "squashed" the decryption function in order to bootstrap, i.e., they published some auxiliary information about the secret key so that we reduce the depth of the decryption circuit to meet this $O(\log \lambda)$ limit. As an example, check section 6.1 of of this paper by Dijk, Gentry, and others.
In Leveled homomorphic encryption schemes, for any $L$, you can choose the parameters in such a way that you can evaluate circuits of depth $L$. As a consequence, you no longer need to squash the decryption circuits, since now, to bootstrap, you can just set the parameters to support a depth $L$ slightly bigger than the decryption depth. That is an advantage, because it means that you don't need to publish information about the secret key, thus, you have less security assumptions. Moreover, public key size decreases, since you are making less things public.
Using the schemes without bootstrapping
LHE are more flexible in the sense that you can use them without bootstrapping targeting some application. I mean, imagine you have an application in mind for which you just need to evaluate circuits with depth $\lambda\log \lambda$. If you use SHE, then you will need to bootstrap anyway, probably to perform the bootstrapping procedure several times after some sequences of operations, but this is very costly! Using LHE, you can set $L = \lambda \log \lambda$, then you don't need to bootstrap and, maybe (depending on the scheme), the evaluation can be much faster.