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We study the representation theory of the type B Schur algebra$\mathcal{L}^n(m)$ with unequal parameters introduced in work of Lai, Nakanoand Xiang. For generic values of $q,Q$, this algebra is semi-simple and Moritaequivalent to the Hecke algebra, but for special values, its category ofmodules is more complicated. We study this representation theory by comparisonwith the cyclotomic $q$-Schur algebra of Dipper, James and Mathas, and use thisto construct a cellular algebra structure on $\mathcal{L}^n(m)$.  This allows us to index the simple $\mathcal{L}^n(m)$-modules as a subset ofthe set of bipartitions of $n$. For $m$ large, this will be all bipartitions of$n$ if and only if $\mathcal{L}^n(m)$ is quasi-hereditary, in which case,$\mathcal{L}^n(m)$ is Morita equivalent to the cyclotomic $q$-Schur algebra. Weprove a modified version of a conjecture of Lai, Nakano and Xiang giving thevalues of $(q,Q)$ where this holds: if $m$ is large and odd, $Q\neq -q^k$ forall $k$ satisfying $\frac{4-n}{2}\leq k<n$; if $m$ is large and even, $Q\neq-q^k$ for all $k$ satisfying $-n<k<n.$ We also prove two strengthenings of thisresult: an indexing of the simple modules when $q$ is not a root of unity, anda characterization of the quasi-hereditary blocks of $\mathcal{L}^n(m)$.

On the representation theory of Schur algebras in type $B$