Maybe going out lets her escape from her mother for a while :) .

My reading has been that she is of a sickly constitution, but it's an interesting idea that it also lets her - and her mother - off the hook for not being accomplished. Lady C is quick to exclaim to Lizzy that 'none of you' draw and only two Bennets play, when as far as we can see Anne does neither. I am sure Lady C would have been sounding Anne's praises if she could - her fine paintings displayed around Rosings and Lizzy would not have been let near the piano if Anne could either sing or play !!

If perhaps her mother had been less overbearing and more gentle, Anne might have been encouraged more. She may, for example, have spent her whole life being told she was too ill to do much, and after a while start to believe it.