'이름'means 'name' so the first sentence can be directly translated into 'My name is Anastasia', and the second is 'I am Anastasia'. But the word '저' is used when you present yourself more politely to strangers or older people. If you dont feel the need to do so you can just say '나는 아나스타샤입니다', which also means I am Anastasia.

It's really just a matter of nuance, and in this case is very similar to the English one: 내 이름은 아나스타샤입니다 = My name is Anastasia 저는 아나스타샤입니다 = I'm Anastasia Just like in English you'd choose between the two based on spoken/written phrase, or the order of introduction between people, so on. Hope this helps!

내이름은아나스타샤입니다 -меня завут Настя 저는 아나스타샤입니다- я настя.