How similar were they in terms of architecture, military, etc.
As promised, here's a more comprehensive answer. I have significantly more knowledge of ancient Indian political history than I do of its artistic history, so I apologize that my remarks on Indian art aren't as well sourced.
Regarding politics, Indian empires have largely been centered on northern India. This is true of Magadha, the Mauryan empire, the Gupta empire, and the kingdom of Harsha. The Mauryan and Gupta empires have both extended far outside of northern India and into the Deccan and beyond. However their control over the south was significantly less direct. In the case of the Mauryan empire, though the edicts of Asoka seem to claim otherwise,^(1) from a logistical standpoint, such direct control would have been more or less impossible, even with infrastructure and technology available to the much later Mughal empire.^(2) In the case of the Gupta empire, they seemed to readily admit that they didn't hold a tight grip over the south. For instance, the pillar inscription of the Gupta king Samudragupta claims that many of the states and peoples in southern India were conquered and then freed.^(3) Why exactly this is is unclear, but if I were to hazard a guess, I'd say the fact that many of the kingdoms of "medieval" India emerged in the south is evidence that there were relatively strong and localized political traditions in the south, making it more difficult to control. Instead, the Gupta kings seem to have exerted influence over the south rather indirectly through political marriages and alliances.^(4) The word for this kind of indirect control is rajamandala (literally translating to "circle of the king"), and a central theme in ancient India's major work of political philosophy, Kautilya's Arthashastra, which dates to the Gupta period.^(5) When the Gupta kings did have to directly intervene in southern affairs, he did so using local custom as the basis for his decisions.^(6)
As far as southern-based states go, there really weren't the same kinds of large, centralized, imperial polities that we see in the north. Your major states here are the Satavahana dynasties and the Tamil kingdoms, but there are several small polities/settlements that began to arise along the lower Krishna River around the 3rd century BCE because of the increased international trade. Here I don't have a ton of specific page references, but an Indologist by the name of Carla Sinopoli argued that the Satavahanas had the "form" of empire more so than the "substance" of empire. By this she means that they exuded an image of a powerful and centralized state, but didn't enact many of the practical policies we'd expect of an empire.
Northern vs southern Indian art and architecture is a whole other topic and it's huge, so I'll be as brief as possible in my remarks here. Generally speaking, there is an intimate relationship between art and religion in all of India. In the north, the clearest examples are the Gandhara-style Buddha from the northwest, and the Mathura-style Buddha from the northeast. The images I've linked are just examples that highlight the differences between the two northern styles--there are plenty more where those came from. The Gandhara Buddha, coming from northwestern India, is highly influence by Greco-Roman sculpture, which you can see in his relative plasticity, his thin clothing, high degree of realism and appreciation of the "ideal" human body. This is because there was a Greek presence still lingering in Gandhara from the conquests of Alexander the Great and the subsequent Greek Seleucid empire. By contrast, the Mathura Buddha is plump, and less attention is paid to rendering his form ideally. In the south, there was great variety in terms of religious art. As Vedism and the Sramana religions made their way into the south, the deities of those religions became conflated with local deities known as yakshas and were incorporated into wider pantheons.
Anyway, I hope this response has been helpful in answering your question. Let me know if I can clarify anything!
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