Well the difference goes some thing like this... (Acutally iam writing this post as a refresher for myself, so anyone else, please add more if you think i have omitted relevant details)
1) A database is an entity to organize data for efficient management. So, in that sense even a flat file is a method of organizing data. But a flat file is not very suitable for organizing very large number of datum. (datum is singular for data which is plural).
For this reason, specialized software which enhance data management came into being.
2) Initial DBMS did not use the ER ( Entity-Relationship Model of E F Codd) as the basis for database management system design.
3) E-R Model of database design based on Relational Algebra was developed by E F Codd. These essentially boiled down to the so called codd's rules for database design. Databases that implemented the ER model of database design are called RDBMS.
However, no commercial database implementation has completely satisfied the ER model completely.
Further topics in this thought process would be things like Relational Algebra, E-R details, DB normalization etc.
If somebody can add further details please do so.