In the past, archaea were classified as bacteria and were called archaebacteria. But it was discovered that archaea have a distinct evolutionary history and biochemistry compared with bacteria.
The similarities are that archaea and eubacteria are single-celled organisms that do not have a nucleus or organelles.
Size and shape
Both archaea and eubacteria are similar in shape and size. They are both found occurring as rods, cocci, spirals, plates, coiled etc.
Difference in Cell structure
The general cell structure of archaea and bacteria are the same but composition and organization of some structures differ in archaea. Similar to bacteria archaea do not have interior membranes but both have cell wall and use flagella to swim. Archaea differ in the fact that their cell wall does not contain peptidoglycan and cell membrane uses ether linked lipids as opposed to ester linked lipids in bacteria.
Archaea flagella evolved from bacterial type IV pili while bacterial flagella evolved from type III secretion system. Bacterial flagellum is like a stalk which is hollow and is assembled by subunits that are free to move up the central pore adding on to tip of flagella while in archaea flagella subunits are added on to the base.
Reproduction and growth
Archaea reproduce asexually by the process of binary fission, budding and fragmentation. Eubacteria reproduce asexually through binary fission, budding, fragmentation, but eubacteria have the unique ability to form spores to remain dormant over years, a trait that is not exhibited by Archaea. Bacteria growth follows in three phases, the lag phase when cells adapt to new environment, log phase marking exponential growth and stationary phase when nutrients get depleted.
Archaea can survive in extreme and harsh environments like hot springs, salt lakes, marshlands, oceans, gut of ruminants and humans. Eubacteria are ubiquitous and are found in soil, hot springs, radioactive waste water, Earth's crust, organic matter, bodies of plants and animals etc.
Video Explaining the Differences