They both have the same meaning. However, the first one sounds a lot more natural to me. The second one sounds a bit strange.
I think it's because "really silly" is an adverbial phrase. You are describing the way that you feel (the verb) with the adverb (really silly). If you split it up it sounds a bit weird.
I guess when you put really before feel you are, perhaps, emphasising 'you' rather than the feeling. I REALLY feel silly, rather than I feel REALLY SILLY. Like you, in particular, feel silly and you want to emphasise that more than how silly (really) you feel.