扰乱 harass; disturb; create confusion; cause havoc with; mess up; perturb
捣乱 to disturb; to look for trouble; to stir up a row; to bother sb intentionally
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扰 in 扰乱 carries the meaning of 'to disturb'; 'to disrupt'
捣 in 捣乱 carries the meaning of 'to smash; to crush'
扰乱(to disturb; to disrupt) is usually in non-physical, non-violent forms
When you talk with your classmate during class, you are disrupting (扰乱) the normal flow of the class
when you look over someone's shoulder while he write, you are disrupting (扰乱) his concentration.
捣乱 (to create a disturbance) is often in physical, violent forms
When you go smashing up a bar, it is an act of 捣乱 (to create a disturbance)
When you trashing someone's house, it is also an act of 捣乱 (to create a disturbance)
Of course your teacher may consider you and your classmate's disrupting action as 捣乱, in a metaphorically sense.
On the other hand, when the police arrest you for smashing up a bar, you cannot say you are merely disrupting (扰乱) its business. It is more serious than just getting drunk and talked loud.
扰乱 is followed by a noun, to elaborate what you are disturbing,
where as 捣乱 doesn't have to be.
小明, 别捣乱。Xiao Ming, Stop messing around.
you can't say
小明, 别扰乱。Xiao Ming, Stop disturbing...., because stop disturbing...what?
So you have to follow it by a noun.
别扰乱别人上课。Stop disturbing the class.
别扰乱我的生活。Stop disturbing my life.
I think this is the biggest difference.
Apart from this, in everyday speaking, 捣乱 is a less serious term, it is usually used when nothing serious happens, i hear this term usually in classes where a teacher asks a student to simply stop messing around or between friends where you just want him to stop playing around when you want to concentrate on something.
Where if you use 扰乱, it means it is a bit more serious, used when the teacher really wants you to stop disturbing the class or you really means he/she is disturbing you on something.