By no means am I an expert, especially on the intricacies of being trans and I don't want to cis-splain their experiences, but I do know that cross dressing has an overlap of necessity with them, but it's a ven-diagram/intersection rather than a circle.
People who are trans typically explain it as having always felt like they've always been in the wrong gendered-body. Wearing clothes that correspond to the gender they feel they are
(also, keep in mind MRI's have shown brain activity that more closely matches the gender they feel they are than the gender they were born with, as well) is way for them to try and reconcile the way they feel with the way they were born. It's usually an early stage in socially transitioning, whether they later choose to chemically/medically transition or not (and keep in mind, this is not always the end goal for people ... simply being acknowledged as the way they see themselves is enough for some people and they shouldn't be afraid to be in public because of that!).
People who cross dress on the other hand may be doing it because they're trans and feeling out the early stages OR ... just because. It could be the way a textile feels, it could be sexual, it could be non-sexual, it could be because they just like it, it could be because ... you see where I'm going with this? There are as many reasons that people cross dress as there t-shirt designs (I may be exaggerating a bit with that one, but you get my meaning, I think).
I know NOFX is problematic, but I feel like "I’m a Transvest‐Lite" explains it decently (for one particular person anyways!):
I'm not transgender, I'm a lazy crossdresser
Who thinks makeup is too much of an ordeal
I paint my toes and wear shiny tight clothes
Not for the look, but how it makes me feel
I don't need things just right, I'm a tranvest-lite
I only shave to do the Time Warp midnight Saturday
I'm done with self-pity, I don't have to feel shitty
'Cause I wanna look pretty, so I give it the old city College try
Don't get me wrong, I still wanna be a guy
Who sometimes likes to dress like a girl
(He sometimes likes to wear diamonds and pearls)
Don't think I don't know I'm not fooling anyone
(He's a cross between Adele and Charlie Chaplin)
You gotta know it's not just girls who just wanna have fun!
Also, the beginning of the song is about the shame felt early on about it and is generally about getting the courage to not give a flying fuck about what people think.
Hopefully I did a decent enough explanation without trampling on anyones lived experiences. The simple truth is that both groups of people have completely individualized experiences and it's a different journey for every single person.