I can kick off the responses with the observation that 'Pax' has much larger blooms and less of them in an inflorescence while 'Prosperity' tends to make more generous trusses of smaller blooms that look very white against the deep green foliage despite the fact that the buds are pinkish and the blooms can be tinged with pink. The blooms are of rosette form and the petals have a lightness that makes the blooms look soft.
'Prosperity' tends to grow as a graceful shrub with long arching canes that are well foliated.
I have not grown 'Pax', but seeing it in other gardens, it did not seem to have that graceful, relaxed shape and the foliage seemed sparser but this may have been because of the way it was pruned.
Need to add a disclaimer that this is based on the roses I know under these names in Western Australia. I'm not sure whether more than one rose is in commerce under either name. or if there are mix-ups in some nurseries or gardens.