Roughly speaking, van Handel is writing from the probabilist's perspective: he spends time discussing "sharper" results such as log-Sobolev inequalities and hypercontractivity. Vershynin is writing from the statistician's perspective.
To make a gross generalization, the results in van Handel are more often useful to a probabilist who seeks to describe the behavior of a very specific, "nice" random process in as much detail as possible than to a statistician who is primarily interested in proving upper bounds under very general conditions.
As further evidence of this, van Handel devotes more time to lower bounds, and the end of the book is devoted to characteristic exponents and cutoff phenomena, which is strictly a probability theory topic--it amounts to describing beautiful mathematical phenomena in great detail.
Vershynin's book, on the other hand, is motivated by applications in statistics and computation. He gives most attention to the tools used most frequently by high-dimensional statisticians.
Statisticians care chiefly about bounding the probabilities of bad events, and accordingly Vershynin's book does not really cover lower bounds, nor does it cover more delicate concentration phenomena such as hypercontractivity.
On the other hand, more time is devoted to the geometry of high-dimensional convex sets and the connection between the the book's results and current work in high-dimensional statistics, machine learning, and algorithms is drawn very clearly.
(Disclaimer: I've read and taken a course that uses Vershynin's notes, and that course listed the van Hendel notes as a reference--I've only skimmed those. However, I do feel confident that the two are different in the way I've described.)