A Burh was essentially a fortified area that was to act as a military strong point that could resist almost anything short of a major attack. They were created, or in many cases designated, by King Alfred to act as part of a larger system that revolved around a system of strong points along the frontier with the Danelaw. The defences at Burhs vary quite a bit since the system as a whole was formed using whatever existing defences were present in addition to new fortifications. These ranged from large towns with pre-existing Roman fortifications such as Winchester and Exeter, to newly constructed fortified centres such as Oxford, Wallingford, Southwark and Cricklade which formed a defensive line in the Thames Valley. At Winchester which had stone walls, we would have likely seen something at least visually similar to late medieval walls. Of note is that this period is in the early medieval period so it is perhaps better to compare them to late medieval walls which are more commonly made of stone. I am not an expert on fortification works, so I am afraid I cannot point out any specific differences between Roman and late medieval walls. Without having seen the show it is also difficult to say how accurate or inaccurate they are visuallly. However, at newly constructed centres such as Oxford, they were defended by earthen ramparts held in by a wooden palisade with a ditch in front of it. Burhs also ranged from town-sized fortified areas like at Oxford, to much smaller areas, what were essentially promontory forts, that seemed to have been designated a Burh. Those Burhs would have been closer to an iron age hill fort in size and defensive capabilities.

The construction of the Burhs and their function as a unified system is attributed to King Alfred's military planning following the Peace of Wedmore he secured with the Danes in 878 AD. There is still some contention as to the Burhs being a systematic fortification effort and to King Alfred's role in their creation. This is in large part due to the scarcity of primary textual sources from the period and the difficulty in interpreting the sources that do exist along with archaeological evidence. We draw the idea of the Burhs being a systematic fortification effort from a document called the Burghal Hidage. It is a document that in essence list out towns and forts along with the hidage (a unit of measure of land). The Burghal Hidage allows us to establish the length of the walls constructed at each Burh based on the hidage assigned to it. In cases like Winchester with pre-existing walls, the ratio of hidage to wall length matches perfectly, while in some other smaller Burhs there are discrepancies in the hidage listed and the length of the walls. There is a remarkable consistency in the type and design of defences constructed at the new burhs, especially, Oxford, Wallingford, Wareham and Cricklade. They are all laid out mimicking the layout of a Roman town with two main roads that cross at the centre of the town. Their similar layout points towards a centrally planned and systematic construction which also occurred around the same time, likely as part of a deliberate system. There was also a second later phase of construction and reinforcement where stone walls were erected, normally on top of the pre-existing ramparts in may of these locations such as at Oxford and Wallingford. However, this would have looked like an earthen retaining wall topped with a line of stone fortifications instead of a full stone wall. In essence, there is sufficient cause to believe that many of the burhs were constructed together within an overall plan or template in mind as the local deviations are not major and could be accounted for due to local conditions such as access to materials or disposition of the builders. 

Due to the nature of warfare where troops were levied, it meant that keeping them in the field for an extended period of time was impractical due to their economic obligations, mostly as farmers. This along with the potential of disease made it difficult to carry out sieges which could be resisted for extremely long times. Most academics argue that Alfred's response to this was to develop his own fortified areas which made attacking Wessex difficult. The burhs being located a days ride away from each other and consisting of strong fortifications and a garrison allowed for the creation of locations were civilians could take refuge in times of crisis, allow them to mutually support each other and enabled them to launch forces to exert a zone of control in their surroundings. They also argue that it could have a wider function to be a military launching off point for invasions since it could also act as a fortified logistics centre and could also have been used to exert control over a newly conquered region.  The relation of the Burh system to the reorganisation of the Fryd, the West Saxon levy militia, is also debated. Other sources like the Anglo-Saxon Chronicle and Asser's Life of Alfred make it difficult to discern the exact link between the two. There are some interpretations that believe that the Fyrd on rotation was to form the garrison of Burhs. This is possible, but given the lack of sources, it seems to be impossible to pin down any exact arrangements. What we do know is that the Burh system did prove effective from the 893 AD entry of the Anglo-Saxon Chronicle which detailed Exeter being able to hold out against a Viking siege. This allowed time for the West Saxons to marshal the Fyrd in other parts of the kingdom and march to Exeter and lift the siege.

On the economic front, Burhs were militarily strategic locations for the same reasons that they were economically important. The most prominent example would be Oxford, which as its name implies was a ford across the Thames. This made it a militarily strategic point as raiders or armies to cross at that location, but also made it an area that saw significant commerce. As safe areas, Burhs were also prime places to become economic centres and this was the case eventually as King Athelstan later decreed that all coins and trade were to be minted and take place within Burhs, demonstrating their later rise to economic importance. However, if their economic value was immediately apparent at the time of their construction or how soon this became apparent is unknown. Although signs point towards the economic role of the Burh not being planned from the outset. There was no primary source from Alfred's era to support the idea that Burhs were built with an economic function. Furthermore, Burhs such as Gloucester, Cricklade and Wallingford faced economic stagnation as Wessex expanded. Some Burhs also never achieved great economic importance at all and did not grow into towns. In this sense, most Burhs would look like a walled town but lack stone walls.

Overall, I would say aside from major towns that date back to being Civitas from the Roman Period, you would not see any Burhs being fortified with a full stone wall. They would mostly be earthwork defences with some stonework added later on if at all. I would hesitate to say they were similar to hillforts since their scale could differ quite a bit and their layout was more Roman derived that anything. But in terms of the composition of their defences, they were much more visually similar to a hillfort compared to a walled late medieval city.

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Sources:

R. Abels, Alfred the Great: War, Kingship and Culture in Anglo-Saxon England (London, 1998).

G. Molyneaux, The formation of the English Kingdom in the Tenth Century (Oxford, 2015).

J. Baker and S. Brookes, Beyond the Burghal Hidage: Anglo-Saxon Civil Defence in the Viking Age (Leiden, 2013)

D.Hill and A.R. Rumble, The Defence of Wessex: The Burghal Hidage and Anglo-Saxon Fortifications (Manchester, 1996)