Discovery in Astronomy vs one in Physics - do they differ in required burden of evidence?
There is no central authority in science. There is no council that sets the standards. The criteria for a discovery are the same: You publish your findings, and your peers accept your results.  

There is the 5 sigma rule in particle physics. Perhaps you were thinking of this. But that is not an official rule, instead it's a convention among particle physics. And it's not applicable to every field. 

Things are never cut and dry. What does "Peers" mean. What if only most are convinced. Does it count? Discoveries are usually judged in retrospect. After the dust has settled and the debate s are over.

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Why is the bar set so high in physics?  When "discovering" a particle there is no way to "see" it, instead you observe a mass of data and see a statistical discrepancy. Cern produces masses and masses of data, and it is sifted for anything unusual. And with so much data, the chance of seeing something unusual is actually quite high. (imagine searching for repetitions of '9' in the digits of pi - if you search far enough you can be sure to find a string of 6 9s even though the chance of a random string having 6 9s is very low)

With a mass of data, and the opportunity to repeat experiments it makes sense to set the bar very high.

Compare with an observation, such as the "chirp" at Ligo. We can't repeat it at will, the data is there: something happened that is consistent with a black hole merger. No other theory has been proposed that can explain the observations. The observation is less dependent on a statistical finding after running multiple experiments, but a single direct observation. 

In fields in which a result could be explained by chance, then a statistical analysis is done, and published. This contributes to the quality of a finding, and so the number of people you will convince. And convincing your peers is the only criteria that counts in the end.