Both are 'Rites of Passage' for almost every pianist at a certain level, and both are extensively overplayed, and badly, far too often. Because of that, even fine performances get on my nerves, just because the works are each horribly over-exposed.
There should be 'nothing soothing' about any part of the 'moonlight' sonata -- the entire work is unsettled, laden with tension and anything but 'sweet.' That it must be said is due to so many badly rendered versions.
Of course you will play each at some point, it is to be hoped the sonata will not be approached until you can handle and play all three movements well, since that is the actual 'piece,' not just one movement, a fragment of the sonata.
You should choose first that which you can learn most readily and play well, always :-)
Best regards.