To an extent it varies by manufacturer which term they use and what period it denotes.
Early Roman and Tullian Roman are probably often coterminous. However, if treated separately, Early Roman would cover the very early history when Rome's army was little more than warbands under a noble commander. The Tullian period would then cover the phalanx warfare at the end of the monarchy and early Republic. Named for Servius Tullius, sixth king of Rome, who supposedly created the centuries which formed the basis for civil and military organisation. Main opponents in this period are other Italian tribes and cities.
The end of this system would be traditionally dated to around 390BC and the Battle of the Allia where Rome's army was annihilated, but as with all stages of evolution the change was probably a bit more gradual.
The "Camillan army" is the first recognisable stage of the triplex acies formation that eventually led to the most recognisable form of the legion. Five lines of infantry, including rorarii and accensi, and leves and velites as skirmishers. Named for Furius Camillus, who supposedly invented it. Opponents would include Italians, Gauls and Greeks, most notably Pyrrhus.
The Polybian army is a later development of, and similar to, the Camillan one but the leves, rorarii and accensi are no longer seen. The two periods may be combined in some lists, as there isn't a clear date for delineating them. The focus is on the hastati, principes and triarii, still deployed in the triplex acies formation. Armies of this period are often depicted with better armour than their Camillan predecessors. Both the Camillan and Polybian armies are supported by native cavalry and substantial allied Italian legions. Named for Polybius, who wrote about it in some detail. Enemies of this period would include Carthage, Numantia, Numidia, Macedon and the Seleucid Empire.
The Marian period starts from roughly the end of the second century BC. The triplex acies maniples disappear and are replaced by uniform cohorts. Velites and native cavalry conventionally disappear too, replaced by foreign auxiliaries. There are some minor changes in equipment from the Polybian period, and equipment becomes more standardised. After 90BC the Italian allied legions are converted to Roman ones (as the Italians are given Roman citizenship). This is the same period sometimes called Late Republican or Caesarian. Named for Marius, who (again) supposedly invented this system). Opponents: Gauls, Pontics, Parthians, Spanish, slave rebellions, Italians, other Romans.
Early Imperial armies are in organisational terms almost identical to Marian ones when it comes to the legionaries themselves. The main differences are in equipment: Imperial legionaries are more often depicted with lorica segmentata (earlier ones have mail) and rectangular shields, among some other differences regarding helmet types. Roman cavalry are reconstituted. Praetorians are introduced. The Auxilia are officially formed, which leads to some greater consistency in the appearance of auxiliaries. Opponents: Germans, Dacians, Britons, other Romans.
The Middle Imperial period would vary somewhat but tends to refer to the period between the death of Trajan (sometimes a later Emperor in the Five-Good- series) and the time of Constantine (or sometimes a little later). Being heavily marked by civil wars, this period saw a decline in organisation and equipment and an increasing barbarian presence in the army. The lorica segmentata fell out of use. Opponents: Sassanids, Goths, other Romans.
Late Imperial covers the last phase of the western Roman army roughly from the Crisis of the Third Century on. Comitatenses and limitanei are the order of the day. The army has changed substantially from that of the late Republic with foederati much in evidence.
"Patrician Roman" is a term employed in some game systems/lists for the very last period of the Late Empire, starting roughly with Flavius Aetius and then continuing with his successors until the deposition of Augustulus, through the Hunnic wars, sack of Rome, and so on.
Opponents in this last period would include Sassanids, Huns, Franks, Goths, Vandals (et al) and, of course, other Romans.