Interestingly as a native English speaker, I would only use "piece of string" in reference to a relatively short cut-off piece from a longer source. On the other hand, if I bought a spool of string at the store, I wouldn't call it "a piece" even if technically it is a smaller division of a longer manufactured string. Once I cut the string, I would refer to them as "pieces" and only then if they were relatively short. To me, "piece" implies something relatively small in comparison to the whole. Other English speakers may use the terms differently, but that is what seems natural to me.
In that sense, the distinction between "string" and "piece of string" is useful in English, and possibly in Swedish as well. "I need a string" sounds weird though, I would only say that if I wanted a guitar string or something, otherwise "some string" or "a piece of string" is a better fit.
Yes, en bit snöre is a cut off bit but ett snöre doesn't have to be. You can't really make the same difference in English for string, but if you'd speak about shoestrings, you could make the same kind of distinction: a shoestring vs a piece of shoestring.
The difference is in how you view it of course, technically you could claim strings are probably usually cut off from longer lengths of string.
It appears that Swedes use "snöre" for the cord that is used for tying shoes: https://sv.wiktionary.org/wiki/skosn%C3%B6re
But "shoestring" is rarely used in this sense in English. You might say you are "living on a shoestring," which conjures up the image of a thin string, a lace that might break, but "shoelaces" are several strings twisted or woven together, sometimes leather strips, for greater strength, and the word "shoelace" emphasizes the function of lacing to secure the shoe: