Willo, IMO Barbossa respects Lizzy for her pirate side, but there's a bonus...she's a gurl. Women in that era were not expected to have a brain and on Barbossa's social strata were not much more than slaves. So for him to run into a fiesty, piratey, brave, young miss from the upper class, probably peaked his interest on more than one level.
IMO as far as Tia needing Jack and Barbossa, she needs someone on her side at the pirate council in order to be set free.