質問: I know that they both mean "finish". But I wonder if there are situations or contexts where you can use one but not the other.
回答: In my mind 「済ませる」 is much more commonly used for negative things. In many cases rather than "finishing something" I'd translate it to "getting something over and done with". For example, the phrase 「宿題を済ませた」 could imply a dislike of homework, or a feeling of inconvenience where 終わる seems more neutral. For another example when paying the bill, 「会計を済ます」 seems to the more common usage. There's also the pattern 「・・・ないで済ませる」to describe getting by without something, where the use of 済ませる emphasises the hardship.