Ah, Mike Love. So, for context, Mike Love was (sort of) the main lead singer (and very occasional saxophonist) in the Californian rock'n'roll band/vocal group formed in the early 1960s called first the Pendletones and then the Beach Boys. Love was the cousin of the Wilson brothers Brian (who played bass), Dennis (drums), and Carl (guitar) (the group also had a second guitarist - initially Al Jardine, and then David Marks, and then Al Jardine again). All members of the Beach Boys sang, the group vocal harmonies being a prominent part of their sound. Mike Love's nasal, conversational vocal style contrasted with Brian Wilson's pure-sounding falsetto, with the two singers being sort of co-lead singers; the typical Beach Boys song would have Love sing the verse melody (e.g., "if everybody had an ocean, across the USA..." in 'Surfin' USA') and Wilson sing the chorus melody (e.g., "Everybody's gone surfin', surfin' USA").
Within the Beach Boys, while Mike Love was sort of the outward face of the band - the extroverted lead singer who was probably the most natural entertainer in the band - Brian Wilson was the leader of the band musically; he initially co-wrote the songs with Mike, with Mike writing the lyrics, and Brian writing the music. By 1963 or so, a couple of years into the Beach Boys' recording career, Brian was also officially producing the music, meaning that he was directing the musicians and studio engineers in the creation of the recorded product (i.e., the role that George Martin had for the Beatles).
The Wilson brothers' overbearing father Murry regularly physically abused his children, and each was marked by this abuse in different ways. One thing likely influenced by this to some extent was that, over the early lifespan of the band, Brian Wilson suffered from mental health problems that would increasingly sit somewhere on a spectrum of mental illness that includes schizophrenia (edit: though not being Wilson's psychiatrist, I'm shouldn't assume a diagnosis), and which his increasing drug usage in the mid-1960s did not help. In 1964, Brian Wilson quit the Beach Boys' fairly incessant touring schedule for mental health reasons, instead focusing on the studio process. This led to an odd situation where he would work on songs in the studio with L.A. session musicians from what later became known as 'Wrecking Crew' such as Hal Blaine and Carole Kaye, and then the Beach Boys would fly in from being on tour to hear their next hit (with Love not being around, Wilson then sought co-writers in the likes of the radio DJ Roger Christian, the advertising writer Tony Asher and then the music scene socialite Van Dyke Parks).
This situation worked, however; the Beach Boys' era of biggest success, commercially, is marked by this division of labour - the Wrecking Crew's versatility and professionalism meant that Brian could create sonic landscapes with emotional heft that the Beach Boys themselves would have struggled to create as quickly and painlessly.
Mike Love, as well as being the lead singer in the Beach Boys, also was the oldest member of the band, being five years older than the youngest Wilson brother Carl Wilson; in 1963, Mike Love turned 22 and Carl Wilson turned 17 (Brian turned 21). Love was also the oldest member of the band in terms of life experience; he was already a married father by the time the band formed in 1961, where the Wilson brothers, with their upbringing, were a bit more sheltered. Love's political and aesthetic preferences are also, shall we say, quite conservative. He was profoundly distrustful of the evolution of pop music through the 1960s in California.
In contrast, Brian Wilson was open to the evolution of pop music through the 1960s, and in fact as a producer of massively successful commercial music who fully intended to keep up with the Joneses/Beatles, he could not afford to ignore it. Thus, Brian Wilson (like the Beatles) played a role somewhere between following the Zeitgeist of the era musically, and expanding the Zeitgeist himself.
So what we have here, in terms of the beginnings of the 'Brian vs Mike' thing in this era is two very different views of what pop music should be; Mike's belief in a formula, formed through observation of what worked with the audiences he played to every day, versus Brian's more progressive studio-based vision. Generally, so long as Brian's vision led to hits, Mike Love grumbled about the difficulties of replicating the sound live (something Brian didn't have to deal with), but didn't make a huge fuss.
However, in 1966-1967 (as fairly well portrayed in the film Love And Mercy), Brian's lovable eccentricities and artistic vision became to look increasingly closer to the proper onset of diagnosable schizophrenia - not that the rest of the band really understood it in those terms at the time.
Firstly, Brian had put much effort in the studio into 1966's Pet Sounds, but this effort - at the time - seemed to mark the point at which question marks started to be raised about Brian. Mike Love, shall we say, did not receive the album purely positively upon hearing it, convincing Brian to change the lyrics in the song 'Hang Onto Your Ego' sufficiently that the title was changed to 'I Know There's An Answer', concerned about the hip L.A. drug slang in the original lyrics. And despite that Brian was focused on the idea of a coherent concept album, in response to his understanding of The Beatles' Rubber Soul, the Beach Boys' record label Capitol did not quite understand the idea of The Beach Boys making a statement album in 1966; they did only the bare minimum to promote Pet Sounds in the US, releasing a greatest hits album only weeks later.
Nonetheless, Brian then went about recording the group's next project SMiLE, working with the counterculture-aligned surrealist lyricist Van Dyke Parks. The record company - after The Beatles' Revolver and then knowing about what the Beatles were about to do with Sgt Peppers - hired Beatles publicity guy Derek Taylor to explain Brian Wilson's genius to the public, priming the public for an odd concept album from a group they might have still mostly associated with surfing and cars and girls. Derek Taylor's publicity for the band is at the heart of the idea that Brian walks upon water, and I gather that Mike Love strongly resented this PR angle as the band's frontman. Additionally, SMiLE marks the point where Brian Wilson's music becomes truly eccentric. Pet Sounds might not have been 'surf, cars and girls', but it was mostly 'girls', albeit from a more adult perspective mixed in with a bit of existentialism that fit right into 1966's general vibe (Brian's use of Tony Asher as lyricist demonstrates the commerciality of the project - Asher being a advertising copywriter). The SMiLE project began with 'Good Vibrations', which really was pop genius and one of the band's biggest hit. But as it went on, the project seemed increasingly...esoteric.
Mike Love's criticism of SMiLE, when he returned home from the band's latest tour, seems to have been more strident than his criticism of Pet Sounds, and it ultimately seems to have been a tipping point that caused the SMiLE project to collapse; Van Dyke Parks, freaked out by Love's negative energy, quit the project after Love asked him what his lyrics mean (in 'Surf's Up', one prominent lyric is "columnated ruins domino"). In Brian Wilson's increasingly fragile mental state, this was enough to push him over the edge into a mental health episode of some sort (involving hearing voices, and some paranoid ideation, whatever the actual diagnosis should have been), bringing the SMiLE project to an end. The rest of the band would release a sort of stopgap re-recorded SMiLE in the album Smiley Smile, and with Brian increasingly out-of-it, the group's focus in the late 1960s become nostalgic songs like 'Do It Again'; Mike Love was increasingly calling the shots.