The connection between Robert Heinlein's Future History and Poul Anderson's Psychotechnic History is that the latter was modeled on the former. The connection between the Psychotechnic History and the Technic History is that both were written by Anderson. Thus, there is no direct connection between the Future History and the Technic History. However, all three are future histories. Heinlein's History and Anderson's second History have in common that, in both cases, the period of time covered by the series as a whole is longer than the life span of any of the characters - although with one exception in the case of Heinlein because one of his science fictional premises is precisely the prolongation of human life. In any case, both series cover a period longer than any expected human life span.
Secondly, as in any fictitious history, earlier stories provide background material and references for later stories so that continuity is provided not by characters but by process. Thirdly, both the Future History and the Technic History follow the same general pattern of capitalist expansion followed by political collapse and tyranny but later followed by a stabler culture.
One pleasure afforded by a future history, as by any long series, is to read or re-read not just individual stories but the entire sequence from beginning to end in chronological order of fictitious events which can differ from the order of writing or publication. With Heinlein and Anderson, there could be the additional pleasure of reading all three histories one after the other in the order of their publication. The sequence within each series is transcended by the sequence between them.
It would be necessary for the reader to remember that, in the case of Anderson, he was moving from very early to much later works by the same author so that a higher quality of writing and imagination is to be found in the Technic History. On the other hand, the Psychotechnic History does successfully emulate the model created by Heinlein and provides a conceptual contrast with the Technic History: a premise of benevolent social control as against unbridled capitalism.
(Addendum: Since writing the above, I have been rightly reminded that, although the social control in the Psychotechnic History is initially "benevolent," it becomes manipulative and oppressive and is overthrown. Consistently throughout his career and in many later works, Anderson passionately defended the uncontrolled and the unpredictable in human affairs and did not want to inhabit any sort of managed or safe environment.)