What is it to be important in this context? Your wording in your side question implies that influence and importance are fairly similar. But of course, a band that only sold 500 copies of an album and didn't really influence anybody might still be an important band for those 500 people. Maybe that band will be re-discovered in 10 years and end up being important to more people.

That said, I think that pretty much on any metric - influence, sales, cultural prominence over a period of time - that you can think of, the Beatles are important to modern music. With the Beatles in 2022, they're also effectively the main product of a multinational corporation (Apple - not the Steve Jobs Apple) who have 55 years of dedicated promotional activities that have been effective in building up and sustaining a brand - recently, the Rockstar games, the Giles Martin remixes of their later catalogue, the Peter Jackson documentary, the Yesterday film, etc. Beatles member Paul McCartney is probably the wealthiest musician currently alive.

Central to the Beatles' 'importance' was the Beatles' enormous popularity in the 1960s in general and especially among young people during 'Beatlemania' in 1963-1964. Famously, they had all five of the top five singles on the US Billboard charts in one week in April 1964, which is a measure of their popularity. This enormous popularity translated into their record company, EMI, making some very healthy profits - it has been suggested (though this is disputed) that one way that EMI used the profits from the Beatles' record sales was to help fund some of the initial development of the CT scan.

If you look at the teenagers in 1963-1964 who were the Beatles' rabid fanbase buying all those records - and the young white Americans and Britons in particular who were a big proportion of that fanbase - you're looking at a pretty classic baby boomer demographic. The baby boomers played a sizeable role in culture once they came of age as consumers in the 1960s, for a few reasons. Firstly, there were a lot of them, demographically - a surprisingly large proportion of the population was baby boomers, because between the Great Depression and World War II, not a lot of people could afford to have many kids. However, post-World War II was a prosperous time with the rise of a middle class, with increasing amounts of disposable income. Additionally, post-World War II was more or less the first time that advertisers (and by extension record companies) started to think about 'generations' like 'baby boomers' as being different demographics with different interests etc, attempting to create an idea of a demographic that can be marketed to.

Music oriented to young people had existed before The Beatles - Elvis Presley had screaming fans in 1956, as the Beatles did in 1963-1964. The American radio DJ Alan Freed is generally most associated with the idea of basically re-branding select proportions of 'R&B' music - music associated with Black audiences in the US at the time - as being music for young white people. But the Beatles represent something of a demographic storm, given the baby boomer demographic coming of age and having money in their pockets in the early 1960s (whereas the smaller amount of 15-year-old teenagers digging Elvis in 1956 were born in 1941, during the war).

So, broadly speaking, you could say that the Beatles are where the record industry becomes irrevocably focused around R&B-influenced pop music for young people in a way they previously had not been; before the Beatles there was resistance to R&B-influenced pop music in the American record industry, but after the Beatles such resistance was futile. Frank Sinatra was a vociferous critic of rock & roll in its early years, but the record company he founded in 1960, Reprise Records, were by 1964 licensing records by the Kinks like 'You Really Got Me' for the US market, and by 1966 had released the 'proto-punk'/psychedelic single 'I Had Too Much To Dream Last Night' by the Electric Prunes. The Beatles made such music acceptable to white audiences by being fairly authentically rock & roll (unlike a lot of crossover pop), but also being palatable - their press conferences were funny, and their public image was, at the standards of the time, a little edgy but basically innocuous. People who weren't inclined to rock & roll found themselves liking the Beatles anyway.

And of course, they were the vanguard of the 'British Invasion' - other British rock acts - The Kinks, The Rolling Stones, Herman's Hermits, Them, Dave Clark Five, The Who, etc - many of which had been signed to record labels in the wake of the Beatles' success in the UK in 1963 - proceeded to dominate the American charts.

The other thing that the Beatles did was, more or less, create the category of the rock band. There were obviously bands that played rock and roll before the Beatles, but many of them were 'singer and basically interchangeable backing band' combinations - Bill Haley and the Comets, for example. Some of those singers were also songwriters (e.g., Chuck Berry or Buddy Holly), largely writing their own material. There were also instrumental bands that weren't simply in the shadow of the lead singer - The Shadows, or The Ventures or The Fireballs - who often wrote their own material. Additionally, there were vocal groups who used a group name and had an unnamed backing band play behind them - The Drifters or The Platters, for example - who rarely wrote their own material (and why would The Drifters write their own material when they had the songwriting team of Lieber and Stoller behind them?).

The Beatles were all of those things at once. They were a vocal group - all members sang and did harmonies - but they were also the backing band at the same time. They were focused around songs they had written. They were not the first band to work in this way - The Crickets, especially when they continued after Buddy Holly's death, fit this mould. The Beach Boys, who had hits with self-penned surf-instrumental-band-crossed-with-vocal-group songs in the US before the Beatles did.

The combination of the Beatles' enormous success and the relative novelty of their approach was extremely influential. Bands that fit most of the criteria were encouraged to fit all of the criteria: up-and-coming London R&B bands like The Who or the Rolling Stones or The Kinks who had made their name around the clubs playing blues and R&B covers basically at some point got told by their managers after the Beatles' success 'okay, you have to write songs now'. By 1967, a group like Brian Poole & The Tremeloes (who Decca signed in 1962 instead of The Beatles) had ditched the lead singer, and were just The Tremeloes, and had a hit with a cover of 'Silence Is Golden'. People probably don't see Metallica or Dream Theater as being particularly influenced by The Beatles - but hey, they're basically guitar-focused self-contained bands that write their own songs, and where the lead singer isn't necessarily always the focus of interest.

If the Beatles had faded away in popularity and gone light entertainment after 1964, they would likely be at the cultural level of someone like Elvis Presley today - which is certainly nothing to sneeze at. Where they transcend Presley in importance and long-lasting influence is because of the 1967 Sgt. Peppers Lonely Hearts Club Band album. This was the biggest selling long player record of the 1960s, and it was indicative of a new way of thinking about recorded music. Before the Beatles, generally, pop music was generally a recording of a performance - a bunch of famous Frank Sinatra recordings of the 1950s were recorded, basically, by Sinatra singing into a stereo microphone with an orchestra playing behind him. The Beatles, on the other hand, by their 1966 album Revolver, had tracks on the album that could not really be played by live musicians using the usual instruments, and which were obviously so - it was the point; they were exploring the possibilities of the medium of the recording. It's a little bit like the difference between a filmed play and a film; just as Citizen Kane marks the point where Hollywood films clearly start to feel like their own medium rather than just theatre plays that have been filmed (and as actors start to act to the camera rather than an actual audience, and change techniques etc as a result), Sgt. Peppers marks the point where the music industry has the capacity to not just record performances of songs, but to create music that is designed to be an auditory artifact, first and foremost, where it doesn't matter if it can be performed live, because it doesn't necessarily ever need to be. The expensive packaging of the Sgt Peppers record reinforces this - it's drawing attention to the record as a physical artifact that is a perfect recreation of a sound created by musicians in a studio.

The sheer success of Sgt Peppers creates a market for this new kind of rock album - this idea that musician was going to release a long-form product every so often that was meant to be a big statement of intent, where sound had been laboriously worked on. It's fair to say that, without Sgt Peppers popularising the concept, the likes of Dark Side Of The Moon or Bat Out Of Hell - big concept records of some sort - would make no sense. Sgt Peppers is not the first album in this sense - Revolver exists, as does, e.g., Freak Out by the Mothers of Invention, or Blonde On Blonde by Bob Dylan - but for better or worse, it's the album that cements the concept. As a result of all the fuss, The Beatles start to become accepted by arbiters of high art culture - you get the prominent composer/conductor Leonard Bernstein rhapsodising on television about the cleverness of the music, and you get high-falutin' classical works like Luciano Beroi's Sinfonia referencing the Beatles in their music.