When a computer is used by many people it is usually necessary to differentiate between the users, for example, so that their private files can be kept private. This is important even if the computer can only be used by a single person at a time, as with most microcomputers.  Thus, each user is given a unique username, and that name is used to log in.
There's more to a user than just a name, however. An account is all the files, resources, and information belonging to one user. The term hints at banks, and in a commercial system each account usually has some money attached to it, and that money vanishes at different speeds depending on how much the user stresses the system. For example, disk space might have a price per megabyte and day, and processing time might have a price per second.
It might be quite embarrassing if my sister could read my love letters.