Actually, they are the same, at least in the same continuity : the hellenistic cataphracts were directly borrowed to the Iranic peoples usages and tied to the presence of an "Iranic front" as with Seleucids.I'm talking about the cataphracts of late Antiquity (as the Romans called them just that), obviously not the earlier ones that go all the way back to the Hellenistic period.
The Iranic presence on Roman borders, with Sarmatians present since the Ist century, allowed heavy-cavalry "standard" presence with Antonines (you can see them on the Trajan Coloumn, and the clientelisation of peoples as Roxolani points this evolution) but the presence of an auxiliary heavy cavalry in the eastern part is well attested, generally (but not systematically) associated with Near-East kingdoms : Armenia mostly, but as well the other all along the border.
Being sort of heavy alae cavalry "pool", it may not been unrelated to the harsh wars that Romans and Persians fought over it (it's interesting to read that some Parthian kings preferred to propose sending Parthian cavalry in support of Romans, as Vespasian, pointing an use of NE cavalry before Antonines).
Eventually, the big difference is more the "decentralisation" of Roman army before the mid Ist century, with more distinct armies due to a very large use of auxilars (especially on cavalry, that was more Celto-German looking in the West, more Helleno-Persian in the East), than a real distinction between cataphractarians and clibinarii (both words being used interchangeably by Romans, with the latter being maybe a bit more informal).
It's not what I said. At the contrary, and I gave the Ostrogothic heavy cavalry as a counter-exemple (I could have used Vandalic cavalry as well), but the point is that most of Germanic cavalry were a Roman/Sarmatic-influenced cavalry with an influence coming back right from the Ist century for Goths (whom ethnogenesis happened at the corner of Romania and Sarmatia).Yes, Germanic cavalry was only slightly influenced back by Roman military developments
As in many, many other features, Barbarians were extremely porous to romanisation, including on military matters (and should I say, precisely on military matters as it was how they were most often integrated on Romania)
One could be tempted to point the use of swords by the Germano-Roman cavalries as a sign of Germanic warfare (and frankly, it would rather be issued from Celto-German warfare as with Gaul heavy cavalry using long swords rather than IInd century's Germanic innovation), but we know that Roman cavalry used it as well than lances...Even if it was a mark of Barbarisation of the army, it eventually lasted as a Romano-Barbarian feature.
If there's an evolution, it might be more on Roman (Byzantine) side than Romano-Barbarian, with a 'reduction" of cavalry equipment in favour of a light polyvalent cavalry approach, and auxiliaries to fill more specialized needs (as Ostrogothic cavalry within Byzantine army).
Conversly, the regionalisation of Frankish armies allows to point the existence of less romanized/sarmatised cavalries (being, litteraly, on the other side of the continent, and in regions where cavalry warfare never played the decisive role it had on Danubian and Mediterranean context).
Not that Alans, Romano-Sarmatian cavalry (influenced or "ethnic") wasn't used by Franks and didn't influenced them : but after the Vth, their use significantly and quickly decline, up to resort to light Alemanic cavalry was some sort of "auxiliary/vassalic".
There the lack of a true cavalry threat, and the quick integration of Gallo-Roman populations within a Frankish identity probably played dialectically.
It's worth noting that Late Roman armies in Gaul, had relatively few cavalry, compared to Britain or Africa, and had to resort on foederati more than in the ERE.
Generally, the WRE had a really fewer cavalry force, while it's clearly obvious on Limitanei (less so on Comitatenses, but these were somehow second-rate; and Scholae).
Not that Frankish horsemanship disappeared : would it be at least for allowing the Carolingian horsemanship to appear, Merovingian cavalry must have been maintained. But it didn't really played that of a significant role (and certainly not a shock cavalry), and may have been more tied to a social status while it survived more importantly in Hispania, and having more Romano-Byzantine influence, which may be the case for Southern Gaul?.
While there's not much indication on this, it's likely that Aquitain/Provencal "Romans" may have kept the heavy cavalry tradition alive, while mixed with Basque (light cavalry, mounted javelineers), Gothic and Italian influence; and eventually the regionalisation must have played a transmission role elsewhere as well.
But, if we go for Western Germanic cavalry per se, as with Alemanic cavalry, it doesn't really looks as either an heavy cavalry, or either a high-status cavalry.
Eventually, the "pattern" develloped there is (and that's valid for a lot of Western European features) a Romano-Barbarian development, a creolisation/mix of both civilisation with an heavy Roman base (and a not-that-traditional Germanic stance).
(I mean, let's face it: Non-mercenary Roman cavalry was always kind of... rubbish. Even during later periods of the empire, when it was much improved since the days of weak cavalry, emblematic of the republican era from several centuries ago.)
I think you're confusing (and you're not the only one) auxiliary and mercenary. Long story short, the former is about military and cultural integration of tactics and equipment. What we call the "Barbarisation" of the Roman Army is actually a Romanisation of Barbarians feature.
Considering, eventually, auxiliaries (or laeti, for that matter) as non-Roman is a bit nonsensical to me. (As in arguing, because Romans had borrowed a lot of Celtic features including for infantry, that Roman infantry sucked while they had to resort to Celts)
Ammianus Marcellinus's fetish for cavalry being a good exemple (even if it does mix heavy cavalry and heavy mounted infantry) on how Romans perfectly integrated for themselves outer features during their campaigns which could be fairly victorious especially when the decisive-battle model, that had an indue publicity and influence on modern tought, didn't really appeared.
Defeats as Adrianople, more due to Roman incoherence than lower quality of its troops, never challenged the importance of Roman cavalry and its fair use (for exemple, among others, Constantine or Julian campaigns)
There was no fluid transmission (the infanterisation of Frankish warfare is obvious and widely accepted) and there's no account I could remember of about a strong medium/heavy middle or late Merovingian Cavalry.Frankish proto-knights of Carolignian times owe the most to influences from existing forms of Germanic medium and heavy cavalry.
Again, not that it disappeared, but it didn't played a decisive or dominant role.
Eventually, it's directly tied to Carolingian, landed aristocracy rise and annexation of peripherical duchies (both last being tied to Carolingian strive for legitimacy and political support base).
The quick transmission of scara as a probable mounted infantry, as an heavy cavalry in Carolingian times hints a more complex pattern.
And giving that the only clearly Germanic cavalry we know from this period, as in either Scandinavian or western Germanic outside Romania mostly used light cavalry if not mounted infantry (it's telling that Saxons were renewed for horse-breeding while not maintaining a huge cavalry, light or heavy); while every exemple of Germanic heavy cavalry is simply too close to Late Roman cavalry for being handwaved...
EDIT : To be honest, RGB seems to make a lot of sense in his last post. I'd tend to think that a mounted archery is less about equipment at hand (late medieval english archery had a lot of non-archery equipment, with two-handed swords, spears, shielf, etc.) than discipline, political/military coherence, and enough training to undergo relatively complex tactical moves and positions, as a predominant use of mounted archery.