I'm asking this question to specifically find out exactly what distinguished these groups from each other in terms of culture aside from difference in language.
More generally I'm curious how different the cultures we assign different labels to actually are from one another? I always read that they are distinct culturally but what does that actually entail?
None really. But that's mostly becauae the "Angles, Saxons, and Jutes" who formed the nucleus of Early Anglo-Saxon identity never really existed in the first place!
But let me back up. The idea that the Angles, Saxons, and Jutes formed distinct ethnic groups who formed distinct kingdoms in modern day England is a fiction, full stop. The peoples who eventually claimed descent from these fictitious tribes were actually of far more diverse origins. The ifluence of Scandinavian, Pictish, Germanic, British, Roman, and Irish influences on material culture, ie dress, jewlery, home building, trade networks, elite culture, weapon styles, and more is quite clear. The people who settled in England during the migratiom period never formed one distinct political unit that could be described as "The Angles" or "The Jutes".
Now historians differ on why this was the case. Peter Heather argues that this was the result of malleable cultureal identity. In the chaotic and dangerous times of Late Antiquity, the various peoples who made up the different ethnic groups that were moving into lowland Britain may have anandoned their previously identities, no matter their origin, in favor of a new identity that was formed in their new political context. So Irishmen, Picts, Norwegians, and Romans all eventually came to call themselves English due to the cultural pressures that were caused by the political breakdown of the Roman world. Malleable tribal identities masked over the actual diversity of these groups, until, in the long run, they formed somewhat cohesive new identities.
However, this view is not universally accepted. Scholars such as Robin Fleming argue instead that the identity if the Angles, Saxons, and Jutes was a later creation born of the increasing centralization of English polities in the early middle ages as an invented source of their legitimacy. This is in turn is predicated on Fleming's argument that the initial migration period was actually rather peaceful, and that only over time did the newcomers rise to the top of English proto states. As new elites came into power they needed to create a story that justified their rule, and thus the conquest narrative of invading Germanic tribes conquering lands fron the British was born.
Now later on as these kingdoms coalesced into the more recognized shapes of kingdoms such as Wessex, Northumbria, Mercia, et al, there were cultural differences. Differences in the worship of pagan gods, differences in language, and more all certainly arose, bit sadly, given our lackluster sources for this time, many of these differences remain unknowable. However, it is worth bearing in mind that these were not inherent differences between the kingdoms and tribes, but rather the result of later cultural shifts that promoted diverse practices.