I've always used "я понимаю" to mean "I understand" but today came across "мне понятно" which, after checking, seems to mean the same. Is there a difference?
Although in some situations these sentences may be used interchangeably, there is a subtle difference.
"Мне понятно" is a neutral impersonal sentence that basically means 'it is clear to me that...' ('A clear picture of the subject is present in my mind'):
Мне непонятно, как ему это удаётся.- 'I can't understand how he manages that.'
Всем ли понятно, что тут написано?- 'Does everyone understand what is written here?'
После твоего объяснения мне всё понятно.- 'Now that you've explained it, it's all clear to me.'
"Я понимаю" can often be used instead of "мне понятно", but the stress here is on my own mental activity:
Я не понимаю, как ему это удаётся.- 'I try to understand how he manages that, but I can't.'
Я тебя не понимаю.-
I fail to understand you (though am eager to).
Правильно ли я понимаю, что...?- 'Do I understand correctly that...?'
Both "понимаю" and "понятно" can be used as replies in a conversation. Both are translated as 'I see', but "понятно" has a hint of finality: 'Now that you told all that to me, it's all clear to me; no further explanation is necessary.'