It's a complex question because we have to separate three things there

- Saxones and Anglii as cultural/territorial/genealogical groups living in Germania during Antiquity
- These groups, among other North Sea peoples which migrated southwards to both banks of the Channel and North Sea
- Emerging political and ethnic groups, following petty-kingdoms lines, in early Medieval England.

They're related, but Saxones and Anglii have different meaning in these circumstances.

Angli was first used by Tacitus to name one of the northern German groups part of the Suevic group, a super-tribal ensemble, barely worth mention giving their small size and peaceful ways for him. It's possible that Aviones, that he lumped together with Angli, were a first mention of Saxones or Axones.Among others ( Varini, Eudoses, Suardones, Nuithones ) they formed a not well known (to Romans) group of small peoples as existed along the Rhine, forming there and there confederations with neighbours, eventually being defined partly trough these relations (as pointed by the name given by Ptolemy, Suevic Angles).

While Saxones had a great historiographical fortune, having formed their own confederation in the IIIrd century similarily to many Rhineland and Germanic peoples did in the same time, Angli seems to have simply vanished probably having been sumbersed by their neighbour's coalitions : Saxones, Eudones, Longobardii, Suevi, etc. There's simply no mention of these in Late Antiquity.

Saxones are recorded raiding Channel coasts, ending up either as repealed or settled on the "Saxon Shore" (possibly both) and they gave their name to the Germanic communities being settled in England, even if other groups are recorded such as Euthiones, having collected a lot of various peoples whom they blended with (people coming from mainland shores of the North Sea, such as Frisians, Franks, etc.) either during the migrations, either once present in England (especially groups coming from more distant regions such as Norway). Anglii are still not recorded in the Vth or VIth century as a distinct or even non-distinct people in England, tough.

Then, mention of Anglii resurface at the least expected place, in Constantinople, in Procopus' accounts of the Wars. He, passing by, mentions Angli (alongside Frisian and Brittons) iving in the isle of Brittia, distinct from Brittania, and having pledged alliegance to Merovingians when they migrated to the mainland.However, these people are hard to place : Anglii in particular are said to be close to Varnes but these lived next to Saxony (altough this might be a remembrance of the aformentioned closeness of Varini and Anglii; or a distinct group of Anglii still present in the mainland...)Brittia had been temptatively located everywhere on the coast between Brittany and the Danish peninsula, so this is not really helpful especially as Procopius is himself skeptical on the story.It's a bit confusing, to say the least, but it points something : first, that Constantinople (and Rome as well from these and other sources) was aware of Merovingian claims of overlordship on Saxons (both continental and insular) which cover some sort of reality for southern-eastern Britain and part of Saxony, altough much much less so than an actual control or vassality (at the possible exception of Kent).Then that either Merovingian or Byzantine courts (maybe both) are still using Tacitus' language to name and label peoples when they need to (Anglii, but as well Eucii from Euthiones, etc.)

In the same time, Pope Gregorius ask to one of his relations to use charity money to help poor peoples or young Angles (quite likely slaves in Gaul) which giving the proximity of several Saxon petty-kingdoms (especially Kent) and the first mention of an English "Angles peoples" set in the Whitby Life of St. Gregory locate them in Kent.

But then, from the 730's or so, Angles are associated with Northumbrians, considering themselves apart from "south-Umberians" kingdoms. It's at this point that the semi-mythified history of Anglo-Saxons began to be set in the stone : but this is not necessarily an ethnic one, even if Bede famously differenciated Angles, Saxons and Jutish kingdoms as beind differenciated by distinct genealogic, royal and migrations in a semi-mythified history.

And then, a different one set by Stephen of Ripon, who makes a distinction between Angle and Saxon, but as he consider the people of Kent as Angles, seems to set their differences as being religious : Angles are Christianized, Saxons aren't or in the process to be. We'd have then a political/religious differenciation : Northumbria and Kent are kingdoms of Angles because they are Christians and dominated by a kingdom of Northumbria tracing its historical origin from Angles already at this point.

(It's to be pointed that when Bede calls the original region of Angles, he calls it Angulus, a word meaning in this period a "desolation" without really associating it to a precise region. He certainly knew about ancient Anglii and their rough location, but couldn't be bothered identifying it while he does with "Old Saxony")

But Saxon remained in large use, partly because of the decline of Northumbria and the struggle in English clergy for regional dominance, set by the rivality of Canterbury and York for the primacy in the island.Stephen argues for a church of the Angles, while Bede supports the claim of Canterbury and the necessity of a separate province of York and set apart a deeper political differenciation between Angles and Saxons along territorial religious and political limits, Mercian charters finally identifying a "Southern Angles" as people under their dominance, under the influence of scholars such as Stephen and Bede, forming a frame for these identities to stabilize.

Basically, Angles and Saxons in Antiquity were different but related peoples from whom we know little apart they were part of the Suevic ensemble.Anglii vanished in Late Antiquity to reappear in relations with Franks, possibly being a not really differenciated people from Saxons (either continental or insular, possibly both), while Saxons were considered as the sole dominant people in England.And then, born out of a double dynamic of mythical reclamation of the past by emerging Anglo-Saxon petty-kingdoms, remembrance or reinterpretation of ancient sources, and political history of the insland, finally formed itself along territorial lines.

Source

Les Angles dans les coins: l’identité ethnique angle en Angleterre du Ve au VIIIesiècle, Magali Coumert.