In my example, the similarity did not require a hard proof but it was not seen for many years
for the reasons which I would call "social".

In 1928 Weil (and simultaneously Siegel) defined and studied heights in algebraic number theory.
In 1933 Henry Cartan introduced the Nevanlinna characteristic of a holomorphic curve in projective space. From certain view point these two things are the same:-)

Only in 1987, Paul Vojta pursued this analogy quite far. This became famous as
"Vojta's analogy", and many new results were proved inspired by this observation.
S. Lang,  was very much excited and widely popularized this
discovery. Since then, Vojta's analogy led to substantial progress in both areas.
The story is described in Lang's book Introduction to Complex hyperbolic spaces, on page 185. 

Why I called the reasons of this almost 60 years gap social? 

Because on my opinion the reason
is that complex analysts and algebraic geometers do not communicate sufficiently with each other.

Actually I noticed the analogy in 1982 (I am sure that I was not alone)
and I told about my observation to a famous algebraic geometer in. He was not excited.
But when the news of the Vojta's analogy reached him few years later, he run into my office,
and said: "Alex, can you quickly tell me what's Nevanlinna theory about?"