Similarities: Both of these religions are linked, and yet fractured, by the tribal nature of their origin. There is no true "Native American religion", nor "Australian Aboriginal religion", insomuch as there are common elements that can be traced back to common ancestors and tribal groups. Neither religion has a "Bible", and what written records do exist are fragmentary in nature; their history was predominately oral.
Some common elements include the deification of natural elements, such as animals or places, the existence of a creator god/gods or some other "origin of mankind", and the belief in the continued existence and/or communication with one's ancestors, as well as the ability of the soul to be reincarnated in a new body. There is a diverse interpretation of "priesthood", ranging from naturally-gifted prophets and seers, to what are commonly called shaman or "wise men", who have some degree of interactive ability with spirits.
There are also some elements in certain myths of both cultures that suggest the mythology predates the "modern" world; for example, stories of submerged forests or great beasts that may represent aspects of the ecosystem which disappeared following the end of the last Ice Age.
Differences: One important difference is the region and peoples involved in this comparison; Australia is considerably smaller than the area populated by Native Americans, particularly if we consider "Native American" to include the Central and South American regions. A major consideration here is the development of writing, masonry, complex mathematics and other rituals among the cultures of Central and South America, which was not seen in Australia, and may be intricately tied to the religion of these civilizations.
A major difference in specific beliefs is the Australian Dreamtime, which is difficult to summarize but functions as a sort of combined spiritual, ancestral and judicial guide for both the living and dead; this concept is common throughout Australian myth.