Wikidata (like most Linked Data services) makes a distinction between the resource itself, and the information describing the resource. In your example, http://www.wikidata.org/prop/direct/P1566
presumably is the identifier of the property itself, whereas https://www.wikidata.org/wiki/Property:P1566
is the identifier for a resource (a wiki page, in this case) that describes the property.
This distinction is important in a linked data context, because linked data (RDF) uses the URI to identify what it is making claims about. So for example, the RDF statement:
http://www.wikidata.org/prop/direct/P1566 rdf:type rdf:Property .
is factually correct, but :
https://www.wikidata.org/wiki/Property:P1566 rdf:type rdf:Property .
is not (it's a wiki page, not a property). Think Magritte: "It's not a pipe, it's a picture of a pipe".
The reason they all redirect to the wiki page in the end is simply that the HTTP content negotiation mechanism makes this happen: your browser sends a request for the resource, with HTTP headers that say what content type(s) it expects, the server inspects that, sees that the client expects HTML, and therefore redirect to the info page about the resource.
In all this, I simply have no idea why there is also a https://www.wikidata.org/entity/P1566
URI, by the way.
More info about how linked data URIs and content negotiation works can be found here: https://www.w3.org/TR/cooluris/ .