Sum and Account have their different meanings as the members have already explained but in modern times these words are being used interchangeably in many situations.
Sum has a plural as Sums and Amount has a plural as Amounts.
In any language the initial intended purpose of the word gets a slow change with time and in a few decades it acquires other meanings also as more and more people start using it that way. In such cases, which are many there, the lexicographers have no other choice except accepting it in its new avatar.
This is the way how many words acquire new meanings in times to come and people use them in that context. Due to these inclusions language always gets modified to that extent.
Knowledge is power.