What you wrote is a common way the two systems are characterized, but it's vastly oversimplified in a couple of ways. First, we need to narrow down when we're looking at since the shogun system was around for about 600 years. Second, we need to define the concept of "state" that we're working with, which in premodern Japan is nowhere near our current concept of "nation state". My answer may be a bit round-about, but it's not a straight-forward concept.

To start with narrowing down when we're talking about, I would love to talk about the whole six hundred years, but my specialty peters out around the 16th century and essentially stops during the 17th. There were definitely changes in political relationships during this time, but I'll have to hope someone who knows more about the Edo period than I comes around and answers that part. For the others, we can break it down into two major sections, Kamakura and Muromachi, and several smaller sub-sections within each of those.

Regarding the Kamakura (1189-1333) and Muromachi (1336-1573) shogunates, these are both two separate military governments - named after where they were located. Within each of these we can break chunks of time down further based on their relationship with the imperial court and relative power, but first I want to go back to the idea of state and say a few words about the court.

Whatever your ideas about states are that at all resemble modern nation states, i.e. defined boundaries, land/human/resource registration, centralized power, generally accepted authority, etc., throw them out when thinking about premodern Japan. They don't apply. Yes, the general physical shape of "Japan" is and was defined by the shape of the islands, but really there could be (and is starting to be) quite a lively debate on whether or not the political powers-that-be constituted something that could be called a "state". We generally talk about spheres of influence, like the imperial court and the warrior government (shogunate). The imperial court was made up of the imperial family and hereditary nobles. Depending on the specific moment, it was headed by either the emperor, a regent from one of the leading aristocratic families, or a retired emperor who was also the head of the imperial family.

The reigning emperor's duties were largely ceremonial, which makes them easy to discount when viewed from the 21st century, but here we have to remove our modern lenses. Ceremony and government went hand-in-hand during the premodern period (any time before 1600, but really even until the 19th century). Any of the contemporary writings about the court or imperial rule inevitably talked about them in terms of either divine sanction, divine succession, or made some connection to prevailing spiritual beliefs. A couple of English language resources here would be Varley's introduction to and translation of Jinnō shōtō ki (by Kitabatake Chikafusa), and Brownlee's book Political Thought in Japanese Historical Writing, which surveys all of the major histories through 1700. Even the word for "governance" in Japanese (政 matsurigoto) is directly related to the word for worship (祭 matsuri, current use means "festival") and I've seen it written in documents using the same character (祭). So the point of this is to show that, even while to us the emperor's role might seem to be just a symbol or a figurehead, to premodern Japanese, the rituals of the court centered around the emperor were the way governance was done.

So what about land management, taxation, etc.? Originally the court based its system on the Tang system in which the emperor owned all the land and granted it piece by piece to individual families, which only lasted a couple hundred years before it broke down, if it ever even spread beyond the capital region. There's no solid evidence that it was widely adopted. Instead, aristocratic families, including the imperial family, and large temples and shrines claimed rights to land. They collected taxes from the capital and rarely, if ever, were physically present on the land. They instead relied on local martial leaders to administer the lands. This, while I'm oversimplifying a bit, is where the shogunate comes in.

The Kamakura shogunate, from its founding in 1189, governed only its own retainers who were the land administrators collecting taxes for absentee aristocrats. While they were't necessarily liked by the emperors (two openly took to arms against Kamakura), the shogunate received its legitimacy from the court, and during the 13th century, several shoguns were picked from aristocratic families. But the shogun also didn't necessarily have direct power over the warrior government, which was instead ruled over by the Hojo family acting as regents for the shoguns. See what I meant by not straight-forward?

So dividing Kamakura into manageable periods, we have the formation of the shogunate, when they received direct support from retired emperors. Just before 1221, certain aristocratic families were increasing support for the shogunate and negotiating sending their sons to act as shogun when the retired emperor Go-Toba essentially declared war on Kamakura and lost. After this, we get a span of time when the Hojo are engaging more directly in court politics, even arbitrating a dispute between two rival imperial lineages. Then in 1331, reigning emperor Go-Daigo called for an overthrow of Kamakura, and eventually some of Kamakura's warriors turned on them and the shogunate collapsed. For a while we have direct imperial rule that ultimately was rather unpopular.

Again, I'm simplifying a bit, but after a while, the Ashikaga set up another shogunate in Muromachi, a part of the capital. For the first 50 years they were tied very closely to one branch of the imperial family, but while that branch legitimized them, we get 60 years of civil war that says that not everyone with political, economic, and martial power felt the same. For one generation, though, during the rule of Ashikaga Yoshimitsu, we do see the shogun acting more like a military dictator and the emperor and imperial court basically does what he tells them to do. He was directly involved in the ritual of the court, which is really important. Some historians go so far as to say that his line could have become emperors had his heir been as successful as he was. After this though, that doesn't happen again. The shoguns, and all the warriors, even the daimyo of the 16th century, get their legitimacy from the imperial court. Even though by that point the court lost much of its economic power, its cultural power was still enormous.

As I said, the political relationship between the imperial court and the Tokugawa shoguns was different, but I don't know enough about it to really discuss it. Hopefully an early modern historian will be able to speak to this for you. To say an answer in one grossly oversimplified sentence, though, the shogun and emperor were heads of two intersecting spheres of authority.

I wrote a lot up there with only a few direct sources, but here are some others that users might find interesting. Kenmu by Andrew Goble, discusses the years around the overthrow of the Kamakura shogunate. The Kamakura Bakufu by Jeffery Mass is a collection of translated documents produced by the Kamakura shogunate that give a detailed picture of what they were involved in. There are also some chapters in Japan Emerging, ed. Karl Friday, that give overviews of the imperial court, the estate system, and the Kamakura and Muromachi governments.

Edit: formatting, fixed a couple of sentences to be clearer.

Key thing to understand here is that the Emperor in Japan is legitimated by the Shinto religion. The Imperial family claims direct descent from the chief diety, Amaterasu. And according to the myths, ruler-ship over Japan is granted to the imperial family by the gods. So this is Japan's version of Mandate of Heaven; and it's an irrevocable mandate.

That explains why the shoguns could not overthrow the Emperor. As long as people believed in Shinto there was no way to replace the imperial family. It was and remains a theocratic institution. Plus according to their myths and legends, the Genji/Minamoto, Ashikaga, and Tokugawa families all claim to be branch families of the imperial family. It's kinda hard for the branch family to replace the main family and still maintain legitimacy.

Now this doesn't mean that there were no attempts to take over or rule outright.

Taira no Masakado claimed to be the New Emperor (of Eastern Japan) being the 5th generation descendant of Emperor Kammu. But this attempt to create a new country in the east failed when Taira no Masakado was killed by his cousin, Taira no Sadamori, and Fuijiwara no Hidesato. There's now a book in English on Masakado's life: The First Samurai: The Life and Legend of the Warrior Rebel, Taira Masakado by Karl Friday.

Ashikaga Yoshimitsu the 3rd Muromachi shogun, became a vassal of the Ming Emperor and thus the King of Japan (note, not the Emperor). Some historians like Imatani Akira think that Yoshimitsu tried to get his second son to be declared the emperor, after he got his wife declared a semi-empress dowager. But this failed because his first son, who was the 4th Muromachi shogun declined to accept the title Retired Emperor for his dead father. There's no book in English on Yoshimitsu's life. Imatani Akira, Buke to Tenno, p14-20.

What there is to 'know' is the reason why this was the case. Why did the Shogun feel the need to retain the Emperor? Why didn't he just depose him? Answering this, you can begin to see how tenuous the power of the Shogun really was. The office of 'Shogun' was merely one of being the most powerful Daimyo, and retaining power was one of the constant concerns of any Shogun. Your power rested on being more militarily powerful than the rest of the Daimyos because you couldn't rely on any sort of traditional loyalty from them. For example, the Ashikaga Shogunate ended principally because the military strength of the family and its supporters was sapped by the invasion of Korea.

The Tokugawa Shogunate attempted to add another layer of control by forcing the prominent families to spend a great deal of their time in the capital. This was a similar strategy to that of the French kings, whose court at Versailles served as a similar control mechanism. Keeping feudal lords away from their power based prevented them from effectively organizing that power into a strong resistance to your rule. The Tokugawa family also married into the Imperial household on a regular basis, but this never made the Tokugawa family more regal, it simply gave another lever of control over that Imperial household.

All this meant that the Shogun's hold on power was extremely tenuous. This helps explain the Sakoku policy instituted by the Shogunate; maintenance of the status quo and preventing any shifts in the balance of power was necessary. Foreign influence, and especially money, were dangerous, particularly given that most foreign trade (and thus wealth) flowed through areas controlled by potential rivals.

It is simple to define the Shogunate as a military dictatorship then, but this is a little anachronistic. It reminds me a lot more of the relationship the Lord Mayors had in Francia during the early Medieval period. It was a feudal system where a single noble house managed to rise above the others and use the office of Emperor as a way to legitimize their rule. Unlike the Carolingians, the Tokugawas were never able to actually depose the monarch (nor is it likely they could have given the much more traditional nature of Japanese society as well as the revered status of the Emperor). This type of position is not that uncommon in the history of feudal states; Charles Martel is one example, but there are many other examples of a single family using their positions of power to dominate the affairs of the state. Japan is somewhat unique though in that it was essentially unthinkable that the Tokugawa family could become the new emperor, and thus their power could never be truly legitimized.