One reason is that probabilists often consider more than one measure on the same space, and then a negligible set for one measure (added in a completion) might be not negligible for the other. The situation becomes more acute when you consider uncountably many different measures (such as the distributions of a Markov process with different starting points.)
Another reason is that probabilists often need to consider projections of events: Instead of asking if Brownian motion (say) has some property at time $t$, we would like to know if there exists a time where Brownian motion has that property. Projections of Borel sets in a Polish space are Analytic (also known as Suslin) sets, and these sets are universally measurable (i.e., measurable in the completion of any Borel measure); a good source for this is [1]. In contrast, projections of Lebesgue measurable sets might fail to be Lebesgue measurable which then hinders further analysis.
[1] Arveson, William. An invitation to C*-algebras. Vol. 39. Springer Science & Business Media, 2012.