You buy the tube to fit the tyre so a tube marked as 700 x 28/38c will be okay for a tyre between 700x28c and 700x38c. Of course in reality it is still okay to use a tube in a tyre which is a bit outside of its rated size but there's no real reason to go too far.
So, yes, using 700 x 28/38c to replace a 700 x 28/37c will likely be fine.
If you want to know about what tyre sizing marks actually mean you could start here.
the 700 number is the wheel size - this doesn't change as your wheels are a fixed size - most road bikes use 700 wheels
the 28/38 is the range of tyre widths that the tube will fit in mm.
so any tyre from 28mm wide to 38mm wide.
is your tyre in that range?
if you replace a 700 x 28/38 with a 700 x 28/37 then only if you had a 700x38mm tyre would you have a problem - and even then it would actually be OK anyway as the numbers are only guidelines and 1mm won't make a difference.
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