What is it about the 20th century music that makes it so different as compared to say, Mozart, Haydn, Beethoven, Brahms, etc? With these composers I feel that there is more "structure" and more of a "tune" then in the 20th century works. Mahler's 9th is quite enjoyable, but I feel as though it's a little "all over the place" and that there's less structure than in the earlier composers I mentioned.
That's a HUGE question

A lot of things make
some 20th-century music sound different: lots of structural, harmonic, and rhythmic ideas came into play that didn't exist (other than maybe in embryo) before 1900. You get complex polyrythms, atonality, bitonality, aleatory (chance) elements, serialism, dodecaphony, spectralism, and on and on. The instrumental palette is also expanded far beyond what Mozart or Haydn knew, so you'll literally hear different sounds.
Be aware, though, that for every avant-garde piece of the 20th-century, you can find one that hews much closer to tradition. You have the witty neo-classicism of Prokofiev's 1st symphony, the high Romanticism of Howard Hanson, the spiritual minimalism of Arvo Pärt, etc. And of course, the early part of the century saw composers born in the 1800's continuing to work in relatively traditional, "accessible" styles: for instance, you had Elgar, Bantock, Stanford, Parry, Bruch, Zemlinsky, Korngold, Atterberg, Sinding, Børresen, etc. (Conversely, you had men like Ives, born well before the turn of the century, going on to write music that still causes people problems.)
There was no switch that was flipped in 1900; you just had a lot of ongoing diversity. (Then again, the 19th century went from Haydn to Wagner, so music got stretched dramatically during those 100 years.)
As for Mahler, he's actually somewhat akin to Mozart in his superabundance of ideas, one coming after another in quick succession. However, that plethora of ideas can work differently in Mahler, namely in that he often juxtaposes themes and tone colors of wildly different character. In the 1st symphony, for example, you get that infamously weird 3rd movement with the lugubrious bass intoning the song we know as "Frere Jacques" in a minor mode, a village band replete with someone banging away on the cymbals, a contemplative lyrical tune based on one of his songs, etc.
But in Mahler, the structure is definitely there, just on a grander scale; plus he often melds the sonata and rondo forms.