Difference between somewhat homomorphic encryption and leveled homomorphic encryption?
Let $\mathcal{C}$ be the set of allowed binary circuits.

Then, a $\mathcal{C}$-evaluation scheme $(\mathsf{Gen}, \mathsf{Enc}, \mathsf{Eval}, \mathsf{Dec})$ that has (i) correct decryption and (ii) correct evaluation is called a somewhat homomorphic encryption scheme (SHE).

A $\mathcal{C}$-evaluation scheme $(\mathsf{Gen}, \mathsf{Enc}, \mathsf{Eval}, \mathsf{Dec})$ is called a levelled homomorphic scheme if it has (i) correct decryption, (ii) correct evaluation, (iii) takes an auxiliary input $d$ to $\mathsf{Gen}$ which specifies the maximum depth of circuits that can be evaluated, (iv) is compact, and finally requires that the length of the output of $\mathsf{Eval}$ does not depend on $d$.

As you can see, levelled homorphic scheme is an "extended" form of SHE. However, if you allow any circuits with depth at most $d$ in $\mathcal{C}$, then you have a levelled fully homomorphic scheme.

If you want to go a step further, there is also $i$-hop and $\infty$-hop correctness for schemes building on top of SHE/[levelled] [fully] homomorphic schemes.

Now let's come to your questions: bounded depth for levelled homomorphic encryption as well as the difference between both should be clear now. For the degree of the polynomial $p$, maybe you were referring to compactness, since it restricts the output of $\mathsf{Eval}$ to at most $p(\lambda)$ bits, where $\lambda$ is the security parameter. I don't know your definition of strong here, therefore I leave it open to you to choose the "better one".

The following paper may help you further in your questions: A Guide to Fully Homomorphic Encryption