Banach spaces $X$ with $\ell_2(X)$ not isomorphic to $L_2([0,1],X)$

You can take any Banach space $X$ for which the weak$^\ast$-dentability index $Dz(X)$ is strictly larger than the Szlenk index $Sz(X)$ (note that we have $Dz(X)\geq Sz(X)$ in general). The reasons for this are that:

1. $Sz(\ell_2(Y))=Sz(Y)$ for any Banach space $Y$ (see Theorem 2.12 of Brooker, Direct sums and the Szlenk index, J. Funct. Anal. 260 (2011) 2222–2246);
2. $Dz(Y)\leq Sz(L_2([0,1],Y))$ for every Banach space $Y$ (see Lemma 1 of G. Lancien A survey on the Szlenk index and some of its applications, Rev. R. Acad. Cien. Serie A. Mat. 100 (1-2) (2006) 209–235); and,
3. the Szlenk index is invariant under isomorphisms.

Indeed, for any such space we have $Sz(\ell_2(X))=Sz(X)<Dz(X)\leq Sz(L_2([0,1],X))$. (As an aside, an earlier paper of Gilles Lancien also compares ordinal indices for a Banach space $Y$ and $L_2([0,1],Y)$; see Lemma 2.6 of Lancien, On uniformly convex and uniformly Kadec-Klee renormings, Serdica Math. J. 21 (1995) 1-18.)

Examples of spaces $X$ for which $Dz(X)>Sz(X)$ include non-superreflexive Banach spaces $X$ with $Sz(X)=\omega$; this is because a Banach space is superreflexive if and only if $Dz(X)\leq\omega$ - see Proposition 2.1 of the aforementioned paper On uniformly convex and uniformly Kadec-Klee renormings by Gilles Lancien, noting also Theorem 3.7 of the same paper which points out - amongst other things - the fact that a Banach space $Y$ is superreflexive if and only if $Sz(L_p([0,1],Y))\leq \omega$ for $1<p<\infty$.

Examples of non-superreflexive spaces $X$ with $Sz(X)=\omega$ include:

1. $X= (\oplus_{n=1}^\infty\ell_1^n)_p$ or $X= (\oplus_{n=1}^\infty\ell_\infty^n)_p$ for $1<p<\infty$, since the Szlenk index of the $\ell_p$-direct sum of a set of finite dimensional spaces has Szlenk index equal to $\omega$.
2. $c_0$ and other isomorphic preduals of $\ell_1$ with Szlenk index equal to $\omega$, such as the Bourgain-Delbaen examples from Bourgain and Delbaen, A class of special $\mathcal{L}_\infty$-spaces, Acta Mathematica 145 (1980), 155-176. Dale Alspach computed the Szlenk indices of the somewhat reflexive Bourgain-Delbaen $\mathcal{L}_\infty$ spaces, showing that their Szlenk indices are equal to $\omega$ in The dual of the Bourgain-Delbaen space, Israel J. Math. 117 (2000) 239-259.
3. The original Tsirelson space $T$ (from Tsirelson's paper) has summable Szlenk index by Proposition 6.7 of Knaust, Odell and Schlumprecht, On Asymptotic Structure, the Szlenk Index and UKK Properties in Banach Spaces, Positivity 3 (1999), 173–199, hence $Sz(T)=\omega$.
4. The quasi-reflexive James space $J$ was shown to have Szlenk index $\omega$ by Gilles Lancien in Indices de Szlenk et espaces de James, Publications Mathématiques de l’Université Paris 6, Séminaire d’initiation à l’Analyse, 1989/90, n 28.

Using similar ideas one can see that you can also take $X=C([0,\alpha])$, where $\alpha$ is an ordinal less than $\omega^{\omega^\omega}$ and $[0,\alpha]$ is equipped with its (compact Hausdorff) order topology. Indeed, it is known that $Sz(C([0,\alpha]))= \omega^{n+1}$, where $n$ is the (unique finite) ordinal satisfying $\omega^{\omega^n}\leq \alpha<\omega^{\omega^{n+1}}$, but for the same $n$ we have $Dz(C([0,\alpha]))= \omega^{1+n+1}$. The computation of the Szlenk index of $C([0,\alpha])$ is due to Christian Samuel (Indice de Szlenk des $C(K)$, Séminaire de Géométrie des Espaces de Banach, Vols.I–II, Publications Mathématiques de l’Université Paris VII, Paris, 1983, pp. 81–91), and the computation of the weak$^\ast$-dentability index is due to P. Hájek, G. Lancien, and A. Procházka (Weak$^\ast$ dentability index of spaces $C ([0, α])$, J. Math. Anal. Appl. 353 (2009) 239–243.)

After this, one can take $c_0(X)$ and $\ell_q(X)$ for $1<q<\infty$ for the various examples $X$ described above (and various other direct sums) to obtain further examples.

I think that one can derive an answer to the question from the main result of Schachermayer, Walter, The Banach-Saks property is not $L_2$-hereditary. Israel J. Math. 40 (1981), no. 3-4, 340–344 (1982), or the further paper Schachermayer, Walter, The class of Banach spaces, which do not have $c_0$ as a spreading model, is not $L_2$-hereditary. Ann. Inst. H. Poincaré Sect. B (N.S.) 19 (1983), no. 1, 1–8.

P.S. The space, let us denote it $X$, constructed by Schachermayer in the first of the mentioned papers provides a desired example because:

(1) $X$ has the Banach-Saks property and hence, by the result of Partington, J. R. On the Banach-Saks property. Math. Proc. Cambridge Philos. Soc. 82 (1977), no. 3, 369-374, the space $\ell_2(X)$ has the Banach-Saks property (Partington proved a more general result).

(2) On the other hand, Schachermayer proves that $L_2(X)$ does not have the Banach-Saks property.

Non-reflexive Grothendieck spaces are examples of spaces for which $\ell_2(X)\not\cong L_2(X)$. The same holds for all $p\in (1,\infty)$.

Particular examples of such spaces include $X=\ell_\infty(\Gamma)$ for any index set $\Gamma$ (and all infinite-dimensional injective Banach spaces) as well as $X=\mathscr{B}(\ell_2)$.

Suppose that $X$ is a Grothendieck space which is not reflexive. For every $p\in [1,\infty)$ the space $L_p([0,1], X)$ is not Grothendieck. This is a result of Díaz; see Theorem 2 in:

S. Díaz, Grothendieck's property in $L_p(X)$, Glasgow Math. J., 37, 3 (1995), 379–382.

On the other hand, the Grothendieck property is preserved by $\ell_p$-sums ($p\in (1,\infty)$).