Let $L$ be the power set lattice ${\cal P}(\{0,1,2\})$. It is clear that there is an order-preserving injective map from $M_3$ into $L$, but no injective lattice homomorphism (because $L$ is distributive, and $M_3$ is not).

What is an example of distributive lattices $K, L$ such that there is an order-preserving injective map from $K$ into $L$, but no injective lattice homomorphism from $K$ into $L$?