I don't exactly know what the Lebesgue sigma-algebra is, but I presume you mean the extension of - for example - the Borel algebra that gives a complete measure. I know this as Baire algebra, and it has a higher cardinality than the Borel algebra.

The $L^p$ spaces however, constist both of equivalence classes of functions, and in fact the spaces are isomorphic via a natural embedding from the Borel one to the other. The difference is that the equivalence classes are bigger. You get more measurable functions in the $\mathcal{L}^p(\lambda)$ space since the sigma-algebra is bigger, but you factor out those you got more when descending to $L^p(\lambda)$.