Yes, preprocessing Beaver triples in an offline phase leads to a faster online phase. The online phase of an AND gate requires just two openings plus local computations.
But there are other advantages as well. Define a "linear representation" $[x]$ to be any way of representing/distributing a value $x$ among parties such that the following properties hold:
The adversary's view of $[x]$ is independent of $x$.
Given $[x]$ and $[y]$, it is possible to compute $[x+y]$ via local computations only.
Given $[x]$ and $c$, it is possible to compute $[cx]$ via local computations only.
There is a protocol that opens $[x]$ to reveal $x$, even in the presence of an adversary (can be formulated for any adversarial model).
For example, Shamir secret shares are one possible linear representation, secure against semi-honest adversaries.
If you assume a setup phase in which parties obtain many Beaver triples $[x],[y],[xy]$ on random inputs, as well as many random $[r]$, then you can do MPC in a very straight-forward way.
This recipe for making an MPC protocol is appealing because you just have to define an appropriate linear representation and plug in. This is what SPDZ, MiniMAC, BDOZ, etc., do (plus other optimizations of course).
In particular, these protocols use a linear representation that is secure against a dishonest majority. If you use computational assumptions during the offline phase, then you can in fact generate such Beaver triples. If you just stick to the VSS paradigm then you cannot get security against dishonest majority.
For a good presentation of this MPC paradigm (abstract linear representations + pre-processed Beaver triples), I recommend the video lectures of Claudio Orlandi & Ivan Damgaard from the 2015 Bar-Ilan winter school.