Don’t Bother Me is notable for being George Harrisons first original song for the Beatles, appearing on their second album With The Beatles.
George Harrison’s first original song, ‘Don’t Bother Me’, was written on tour in August 1963.

In my view it is better than several of Lennon-McCartney songs (in this ranking it is higher than Little Child, I Wanna Be Your Man and Love Me Do), so the possibility was there from near the outset of their recording career for George to develop a strong contribution as a songwriter.
The Lennon-McCartney partnership was more productive and more innovative, but that is surely partly because two heads are much better than one. As well as contributing directly to specific songs, they could help one another develop: sharing and refining critical and creative principles. And this process is cumulative, so they were not merely running faster, but accelerating.
Still George eventually caught up.
Don’t Bother Me shows already some of the traits that became part of his distinctive style: a different approach to chords which he tended to combine in original but not alarming ways. I am no music theorist but this song which is transcribed in the key of Em contains (prominently at the beginning of each line of the verse) B7 and E7 which don’t naturally fit. Apparently these can be interpreted as “Pentatonic and Dorian inflections“.

It’s my impression that you’ll find a little harmonic weirdness in most of Harrison’s songs. I think he started off doing this to try to keep things original, stuck with it because he didn’t want to be too obvious and later developed this instinct further e.g., via the different rules of Indian music.
Don’t Bother Me also has a pretty negative lyric: “it’s not the same, but I’m to blame”, “go away, leave me alone, don’t bother me”, which came to be another Harrison trademark. Later examples include I Me Mine and Wah-Wah, although he eventually moved (intermittently) in a more positive spiritual direction (Here Comes The Sun, My Sweet Lord). The idea that George wrote negative songs is a little bit of a cliché, but it has does have some truth I think.
I think both the distinctive harmonic approach and the willingness to address negative feelings can, in the right context, work by complementing or offsetting some of the more harmonically straightforward and lyrically positive songs that make an album accessible and popular.
One of the things I like about the Beatles is the variety of voices, musical and lyrical ideas they put together in a single album, and I think Don’t Bother Me provides some of this variety on With The Beatles.
Maybe the other Beatles could have done more to improve the arrangement – although there’s some additional percussion, there are no vocal harmonies or answering phrases, and something along those lines might have made the song more compelling.