Why didn't he get the power of Mjolnir?

Let me counter this with another question (and an admittedly slightly speculative answer): What makes you think Vision didn't get the power of Thor at all? In the movie we see Vision actively wield Mjölnir, not only when he hands it to Thor, he also uses it in combat. So he definitely can wield the hammer and make use of its power. The question remains then, what you define as "the power of Thor" then (and how much of that is actually related to the hammer directly or more to Thor). Is it the lightnings? We don't see him use them but he might just not have made use of them in the few times he wielded the hammer. Is it the strength and invulnerability? Well, Vision had that already anyway, given his perfect bio-mechanical bodily form.

Or is it the ability to rule over Asgard? I would say Vision is actually worthy of that, too. As described in my answer to another highly related question Vision is actually worthy of the hammer and scenes with him carrying the hammer a supposed to convey exactly that:

Vision seems to be the embodiment of the ancient ideal of a perfect mind in a perfect body, one that combines the advantages of a mechanical intelligence with the concerns and soul of an organic lifeform and brings compassion for his (undoubtedly inferior) human fellows, in contrast to his purely mechanical brother Ultron. He thus seems to be the ideal creation of a humanity that strives for posthuman evolution. One could say in his second try Tony Stark (with the help of an actual (demi)god and some magic alien artifacts, though) overcame his role as merely a "Modern Prometheus" and didn't just create sentient life, but a new god in itself.

And the fact that Vision is able to wield Mjölnir to me doesn't come from the fact that he is "just a machine", as Tony and Steve argue jokingly at the end, but from the fact that he is superior to us all not only in his body but even more so in his mind. And this I think is the attitude that Mjölnir is attracted to, as already shown in the first Thor movie and also in this one when Steve nearly achieves to lift it (which was not because he is the strongest, but because he is the noblest at heart). In fact, Tony's and Steve's theories about Vision being "only a machine" at the end even support the interpretation that this is not the case by the mere humorous way they are presented in. Of course their childish excuses are not the reason for why The Vision achieved what they couldn't. In fact his pickung up the hammer is a perfect and to the point illustration of Vision's superiority and immediately clarifies that to the viewers, a symbol that has been explicitly set up by the earlier scene (with the whole gang trying to pick up the hammer) for exactly that purpose.

So yes, Vision does get the power of Mjölnir and Thor with everything connected, including the abstract notion of "being worthy". But what makes Vision so extraordinary as part of his "worthiness" and what makes this worthiness and power seem so trivial (without the nice effects described in the other answer) is that he doesn't strive for this, he doesn't care to rule over Asgard, not because he can't as "a machine" or because he doesn't have a soul, but because his mind and motivations are above such "worldly" considerations. His light-handed and downplayed way to hand over the hammer to Thor and his easy ability to carry it in battle convey both his worthiness and his not caring about this at all. He is a god without a god-complex.

So I'd wonder if there's anything else that defines "the power of Thor" and that Vision lacked or didn't demonstrate. The fact that we don't see him use all of Thor's powers to the fullest might just be due to the fact that Vision doesn't need to use them, he has his own powers (which aren't inferior to Thor's at all). And using Thor's or Mjölnir's powers above being necessary is not in his nature, as explained.

As Steve and Tony try to comfort Thor at the end of Avengers: Age of Ultron, they mention that an elevator can lift Mjolnir, but an elevator is not worthy. We see this example in action in both The Avengers (when the Helicarrier is capable of lifting Mjolnir) and Thor: The Dark World (when a coat hanger is capable of lifting Mjolnir), so we know that it has at least some basis in the reality of the Marvel Cinematic Universe.

It would seem that as The Vision is an android, he is exempt from the rules of Mjolnir, meaning that he can lift it, but he is not granted the power of Thor. As I have said in this answer, when we do see The Vision use Mjolnir rather than simply holding it (when he hits Ultron with it), we see that he hits no harder when 'wielding' Mjolnir than he would have anyway.

Additionally, when we see Thor regain his worthiness in Thor, his lifting of Mjolnir is accompanied by various special effects indicating him being 'powered up', none of which are present when The Vision lifts Mjolnir - because he isn't worthy, and has not been granted the power of Thor.

In short, The Vision can only lift Mjolnir because his status as an android would seem to make him exempt from the rules, meaning that he is not granted the power of Thor.

There is only one scene where Vision uses the hammer. He swings it, Ultron goes flying and then he gives the hammer back. What kinda power transfer are we talking about here? Did you expect Vision to sprout blond hair too? It's a weapon with cool powers. Who ever holds it, uses the powers of the hammer, they don't get infused with it. Otherwise everybody who holds the hammer would be Thor.