"Caroline, No" (The Beach Boys) & "A Day in the Life" (The Beatles)

So it's common knowledge that The Beatles' Sgt. Pepper's Lonely Hearts Club Band and--to a somewhat lesser extent--Revolver were influenced by The Beach Boys' album Pet Sounds, which itself was influenced by The Beatles' Rubber Soul.  
The Beach Boys' resident genius and primary songwriter Brian Wilson had been impressed by the cohesive nature of 1965's Rubber Soul, considering that most pop albums of the day were typically a loose collection of singles and "filler," slapped together by record company execs without any real regard to continuity.  (Although, it could be argued that the American version of Rubber Soul still bore the heavy-handed imprint of Capitol execs, who replaced 4 songs from the official British release with selections from the Help! soundtrack to cash in on the growing popularity of folk-rock in the States...)  
In any case, Wilson sensed that Rubber Soul had exploded the boundaries of what rock and pop music could be: The Beatles had written every song themselves, played every note themselves, and had expanded the sonic palette of pop music, throwing in everything from sitars to Baroque piano.  It triggered such a reaction in Wilson that he immediately set out to create "the greatest rock album ever made."  Thus commenced the great creative rivalry between The Beach Boys and The Beatles from 1965-7, each trying to best the other on the charts while also racing to the top of a creative Everest.  
Paul McCartney and John Lennon heard Pet Sounds for the first time at a party in London in 1966, where The Beach Boys' Bruce Johnston (who took over Brian Wilson's vocal and bass-playing duties on the road when Wilson stopped touring in 1965) played the album for them in its entirety.  Twice.  The album instantly inspired McCartney to create "Here, There, and Everywhere," a track found on Revolver that pays tribute to Wilson's compositional style and The Beach Boys' close vocal harmonies. 
Fast on the heels of Revolver's release, The Beach Boys' single "Good Vibrations" hit the airwaves in October 1966, followed in February 1967 by The Beatles' single "Strawberry Fields Forever."  Although Wilson's mental state was already fragile--largely the result of record company pressures, creative disagreements within the band, increasing drug use, and his own perfectionism/self-doubt, "Strawberry Fields Forever" might have been final coffin nail.   
In Director David Leaf's 2004 documentary Beautiful Dreamer, Wilson's friend Michael Vosse recounts riding in Wilson's car in early 1967 and hearing "Strawberry Fields Forever" on the radio for the first time.  The song so affected Wilson that he pulled the car to the side of the road and rather cryptically said, "They did it all already.  They did what I wanted to do with SMiLE."
Ultimately, SMiLE was shelved until 2004, and The Beach Boys/Beatles rivalry came to a close with the final, thundering piano chord at the end of "A Day in the Life."

So, with that, I present "A Day in the Life" and "Caroline, No" for your consideration.
There are a few small similarities between the two songs.  In addition to being the final track on each of their respective albums, both feature observational lyrics about relatable aspects of life: Lennon/McCartney wryly recounting headlines from the newspaper, which range from shocking to the mundane; Wilson/Asher exploring the topic of running into an old flame after many years, only to find that she has lost her spark and innocence.  Then there are the sound effects that close each track: snippets of random Beatle chatter, chopped and spliced into a mishmash of noise, running ad infinitum in the runout groove; Wilson's dogs, Banana and Louie, barking at the sound of a passing train.
In other aspects, the songs are vast worlds apart.  "A Day in the Life" is epic, dramatic, and orgasmic.  Its gradual sonic swell and Lennon's cheeky sex/drugs double-entendre, expressing his desire to "turn on" anyone within earshot before the whole affair erupts in a cacophony of horns and strings, is the aural equivalent of stoned coitus in the middle of Piccadilly Circus at rush hour.
"Caroline, No," on the other hand, is a guy sitting in his dusty attic, leafing through an old yearbook with a dim flashlight that's about to run out of battery power.  In the way "A Day in the Life" beckons the world to tune in and turn on, "Caroline, No" quietly asks the world to go away.  It's an intimate lament for fleeting beauty.
In many ways, I think these 2 songs underpin the psyche of the late 1960s: a simultaneous desire to shed the old while also pining for innocence and simplicity of bygone days.  (From a musical standpoint, it's that bi-directional pull that eventually spawned the heavy/light dichotomy that is heard on The Beatles' "white album" and that characterized the entire aesthete of bands like Led Zeppelin at the turn of the decade.)