Both species belong to the same family, Acanthiza, look quite similar, but behave quite differently. Yellow rumps feed mostly on the ground, and the Browns mostly in shrubs and trees, sometimes in the outer foliage at considerable height. Despite that, their paths do occasionally cross, m
I believe the numbers of Yellow rumps have declined over the last twenty years, at least in the South of the state. Bob Green, writing in his "Birds of Tasmania", describes them as "uncommon and sedentary". My impression is that although they appear sedentary during much of the Spring and Summer, for the rest of the year they appear to be nomadic. The Risdon Brook Park is one local area where they could be described as common, particularly among the grassland and decaying wattle scrub.
Brown Thornbills frequently bathe, but I have yet to see the Yellow rumps do so, even when they have had the opportunity. Brown Thornbills build a domed nest close to the ground, and I have found most nests in low scrub and ferns, often after accidentally flushing the sitting bird. Yellow rumps nest in the lower branches of shrubs, such as acacias, often when they are in flower. I've often wondered whether the choice of yellow flowering shrubs to nest in was by design or chance. Yellow-throated Honeyeaters seem to do similarly. The Yellow rump's nest is something of an enigma. It's a hanging domed arrangement with the entrance near the base, but on top of the dome is a normal open "conventional" nest. It's often suggested that this is a ruse to confuse predators, which it may well do, but I suspect that there may be some other explanation as to why it's there. Browns are agile acrobats, frequently feeding while hanging upside down (as in the lower image), something I've rarely seen Yellowrumps do, they are more often encountered when flushed from grassland, when their yellow rumps make identification easy.
By the way, I'm still trying to get good images of the Yellow rumps!
[Images: Top, Yellow-rumped Thornbill, bottom, Brown Thornbill]