In this context, both 明白 and 懂 have the same meaning of "understanding". When YOU apply the words in your own speech, the difference is that 懂 can be further applied to describe what you know (knowledge) or what skills you have. E.g. you can say 我懂英語 which means "I know English" (whereas when you say 我明白英語 it means "I understand how English works", which is a bit awkward we seldom say things like that), or for skills e.g. 我懂得使用Photoshop, "I know how to use Photoshop", whereas you cannot say 我明白使用Photoshop

I've always considered 明白 to be more like 'comprehend'.