I use the word 'sympathise' much more. The word 'pity' carries more weight and meaning to me. I tend to use 'sympathise' when thinking objectively, and 'pity' is usually a word I use inwardly to describe a feeling I have; Sorrow for somebody else.
To expand, 'sympathise' is used commonly and in all sorts of contexts on a daily basis. From 'customer service' and business tactics, to intellectual debates. "I can sympathise with that, but.."
Whereas if I say 'pity' out loud (rarely) it's usually passively aggressive or comical; 'I pity you', 'I pity the fool', .. Thanks to a few modern movies promoting it's usage in commical/dramatical scripts.
But sometimes, when I'm searching through my feelings and for a name for them them..genuine 'pity' comes to mind. Thinking about the atrocities we view on the news everyday is a good example.
And yes, I suppose the word 'pity' does come hand in hand with some notion that you are better of than the subject which you are 'pitying'.
And it's that realisation that is humbling and makes up stop and reflect. Hence why , we shy away from assessing our 'pity' and instead occupy our selves with our 'sympathies'.