The simple answers is yes, they're synonyms. So you're right in suggesting they are the same thing.
However, the more complicated answer is that whenever you look at a list of possible synonyms and decide which one to choose there is often a deeper process connected to how informal/formal, and how commonly used, a word is. I wouldn't say that 'achievement' is less formal, but it is more commonly used than 'accomplishment', therefore the decision to choose the word 'accomplishment' possibly suggests a bit more care has gone into choosing the word, and this choice possibly reflects the sense that an 'accomplishment' has a greater value. So it's a bit of a 'grey area', but on the whole I'd say 'accomplishments' are worth more than 'achievements'.