Tl;dr: Shinto today structurally more closely resembles the buildup and remains of State Shinto on an organizational level, though culturally and ideologically it is closer to its pre-Meiji form. Altogether, it is a natural evolution that takes on its own identity in Japanese day-to-day life. 

What eventually became Pacific War-era State Shinto was born around the time of the reign of Emperor Meiji in the mid to late 1800s, out of a desire to reclaim "Japanese-ness" and expel foreign influence. In a religious context, the first step here was shinbutsu-bunri, the formal separation of Shinto and Buddhism. For centuries before this point, many Shinto shrines and Buddhist temples had coalesced into independent, shared complexes with shared income, so this forced (and in some cases, violent) separation of the two was an incredible financial and organizational detriment on both sides. To counter-act this loss, the state began to appoint its own priests and caretakers of shrines, as well as providing financial support and attempting to unify the web of shrines scattered throughout Japan, numbering in the tens of thousands. All of these elements came together to support the (from a historical perspective) rapid indoctrination of Shinto under an authoritarian government.

Now that the government itself controlled the shrine system, they began the process of filtering out what was and was not useful to them. This led to the birth of existing beliefs being classified into kyoha (Sects), while Shrine Shinto, what the government was calling its own brand, became less spiritual and more ideological. The focus shifted from a more personal experience to a state-required activity. Shrines became hubs to indoctrinate the populace with Kokutai. Kokutai, roughly translated as national essence, national polity, national identity, etc., is core to State Shinto and revolves around imperial reverence and theological ultranationalism; in a sense, the worship of the state itself, and, through the authoritarian regimes that followed, the protection and expansion of the state. Up to this point, the imperial line had been universally acknowledged as descendant of divinity, and their word was law, but never to such a widespread, authoritarian extent.

Kokutai was the iron soul of State Shinto, and also the first thing to be abolished by the American occupation in the post-World War 2 period. The broadcast of the Emperor decrying his own divinity was only the start. The separation of the shrine system from the government was the other major change. To this day, shrines are still disallowed from federal funding. In this way there was almost a reversion to pre-Meiji shrine policy, though they still remain separate from Buddhist temples. The prominence of the Emperor has also remained greatly diminished politically and theologically, though culturally it's probably fair to say he's a bit more important to Japanese than, say, the Queen of England to the English.

Outside of this...much of Shinto has been almost irreversibly changed from Shinbutsu-bunri and onwards. Although much of the nationalistic fervor has left the belief and Shinto has gained back its spiritual elements, it is still widely considered (both inside and outside of Japan) inherently Japanese, and a critical part of the Japanese identity. The range between cultural practice and religious belief varies from person to person, though the general opinion seems to be that Shinto today is more of a cultural tradition than anything else. 

Organizationally, many of the shrines included within the State Shinto system were shuffled into the modern Jinja Honcho, as close to a central authority as Shinto has today. With no government funding yet a more unified structure, Shrine organization today is neither pre-Meiji nor State Shinto, but a new hierarchy. Major holidays and festivals remain events of national importance in which millions of people flock to shrines to celebrate or participate in, but these are no longer events of ultranationalist importance, but rather closer to celebrations like Christmas, Thanksgiving, or New Years. 

Politically, many right-wing or extremely conservative parties may cling to State Shinto or other tatters of the war years as relics of a "greater time" or "the height of Japanese power." Yasukuni Shrine, where war criminals and war slaves alike are enshrined, specifically stands as an ongoing political issue in the region for a wide variety of reasons. Regardless, Shinto itself in its modern form does not reflect these nationalist values, but is rather used as a crutch for a wide variety of political ideologies across the spectrum.

Ideologically and theologically, Shinto today is proud of its ancient heritage and upholds those rituals, values, and concepts very well, without falling to becoming a nationalist tool. Many shrines, regardless of Shinbutsu-bunri, have a lineage leading back hundreds of years. Many of the core ideas, including the purity of humanity, the nature of kami and divinity at large, reverence and respect for nature (both in beauty and in disaster), and ancestral worship, still exist and have been largely unchanged over the centuries. All in all, Shinto today is a much better and much more respectful form of the tradition than the State Shinto form which exposed it to the world at large during the Pacific War.