According to my knowledge so far:
- だんだん indicates that a state changes gradually or little by little.
- ますます indicates that something happens to an even greater (or lesser) degree than before.
That is to say, in だんだん the focus is on the gradual change from one state to another, whereas in ますます there is no change of state, but the degree intensifies (or abates).
Now, the distinction seems clear at first, but I've come across an exercise where I'm to choose between だんだん and ますます to fill the blank in some given sentences and I can't figure which is the right choice. This is the exercise:
- 紹介された時は変わった人だと思ったが、何度か会って話している間に （だんだん/ますます） いい人だと思うようになった。
- この映画は、最初は面白かったが、（だんだん/ますます） つまらなくなっていった。
At 1. I'd go with だんだん, because there is a change of the opinion (from "this person is different" to "this person is good") on that person, and it is a gradual change. However, we can also say that the speaker is "more and more" convinced of the good nature of that person as he meets and talks to him. So I'm not really sure if the answer is ますます (or if both options can be chosen).
At 2. I'm inclined to choose だんだん as well, because there is a change of state (from interesting to boring), and this change of state happens gradually as the movie moves forward. However, reading this sentence I can relate to the idea of a book or movie "growing boring", "more and more boring" (until you get fed up and turn off the TV or change the channel). If this is the intended meaning, maybe ますます is a better choice. Moreover, if the sentence started like この映画は最初はちょっとつまらなかったが、見続けようとして、（だんだん/ますます） つまらなくなっていった, i.e. the movie was already boring from the beginning, ますます would be a better choice because there is no change of state, just a change of degree (of boredom).
In other words, at 2. I don't know if the intended meaning is "the movie was interesting at first, but it gradually became boring" (it changed gradually until at some point you found it boring, だんだん) or "the movie was interesting at first, but it became more and more boring" (first a little boring, after a while quite boring, and eventually unbearably boring; the more you watched, the boring it became, ますます).
As you see, I am aware of (some of) the differences between だんだん and ますます but I am very unsure of how to actually use them.
Note that there are other words and sentences (5 words and 5 sentences) in the original exercise that I have excluded because they weren't relevant. As the amount of word choices matches the total amount of sentences, I'd expect each of the 5 words to be used in one of the 5 sentences, but I chose だんだん twice and I didn't pick any ますます. Therefore I'm pretty sure I've got at least one of them wrong.
- Is there any other difference or nuance between だんだん and ますます that I am not aware of?
As per the sentences 1. and 2.,
- What is the right choice in each case and why?
- Can both ますます and だんだん be a valid choice or there is unequivocally a single right choice for each sentence?
Note that in this Q&A, だんだん is used to translate "better and better", which AFAIK it would be more appropriately translated as ますます ("more and more") than だんだん ("gradually more").