I don't know if this is what you have in mind, but to me, these poems have very smiliar themes. They are both about how true love does not try to force the loved one to be some certain way.
In Sonnet 116, Shakespeare talks about how love does not seek to alter the person it loves. It says that love does not really care about what the person looks like -- because that changes over time.
In Sonnet 130, Shakespeare is talking about the imperfections in his love. He is rejecting the idea that one's love must be perfect (sort of like how people today should reject the idea that women must all look like supermodels). Instead, he is saying that she is just as lovable as someone else who (people say) is perfect.
So both poems are about true love being happy with what it finds -- not wanting a perfect lover.
Sonnet 116 is about the nature of love and claims that love holds fast and is not shaken by troubles and tribulations ("tempests"). Shakespeare claims that love that is shaken and turned to loss of love, was never love at all" "Love is not love / Which alters...." He ends with the emphasis of an ironic twist saying that if what he says is not true then it is also true that no man ever loved and that he never wrote anything. Shakespeare proves this thesis in Sonnet 130.
Sonnet 130 is Shakespeare's praise to a beloved who is apparently anything but fair, e.g., wires for hair, no roses in the cheeks. Yet Shakespeare speaks of a love as true as any for a beautiful woman. He states that he thinks his love for this unlovely woman is as "rare," meaning excellent, admirable, fine (Dictionary.com), as any bespoken for a beautiful woman.