Objective: DBMS VS RDBMS:

DBMS : dBase, Access, IBM's IMS
RDBMS : Oracle, PostgreSQL, SQL Server, MySQL, IBM DB2

A RDBMS uses inter-related collections of Tables, and each table has a reference point to, one or more, other tables.

A DBMS is used to store data in files, some relations among the files can be established. A RDBMS is a DBMS which stores data in tables

DBMS can be used for simple applications whereas its next generation design i.e. RDBMS can be used for complex business applications. IBM's Information Management System (IMS) is an example of DBMS and Microsoft SQL Server is an example of RDBMS.

RDBMS imposes integrity constraints (ACID) and  normalization of tables, and DBMS does not support either of these. 

DBMS contains flat data, whereas RDBMS maintains some relation between the entities.

DBMS support single user, and RDBMS supports multiple users.

DBMS used for small amount of data, slows down with complex and large amount of data,  RDBMS is comparatively faster than DBMS.

Data redundancy is common in DBMS, with key-constraints and indexes  in  tables RDBMS avoids the data redundancies.

Dr.E.F.Codd proposed the following rules for a DBMS to qualify as a RDBMS; however the rules 6,9,10,11,12 are difficult to satisfy.

Rule 0: Foundation Rule: The system must fulfill the condition of being relational as a database and also as a management system.

Rule 1: The Information Rule: All information must be represented by values in column positions.

Rule 2: The guaranteed Access Rule : Each and every scalar value must be accessible using the table's and/or column name, also columns of related tables using Primary key.

Rule 3: NULL Values: Must allow a field to have Null (or empty) values.

Rule 4: Active online relational  must be accessible to authorized users.

Rule 5: Must support at least one language comprehensive in supporting data definition, view definition, data manipulation, security and integrity constraints, authorization and transaction boundaries.

Rule 6:The view updating Rule:  System must be support updating the theoretically updateable views.

Rule 7: Data manipulation Rule: System must support High level operations like insert, update and delete.

Rule 8: Physical Data Independence: System must possess physical data (arrays, collections) independence.

Rule 9: Logical Data Independence: System must possess logical data (table, columns, rows) independence.

Rule 10: Integrity constraints must be specified separately from application programs, and possible to change if required.

Rule 11:The distribution of portions of the database to different locations should be invisible to the users.

Rule 12: A low level language (single-record-at-a-time) can't not be used to subvert or bypass the integrity rules and constraints that are expressed in higher level(multiple-record-at-a-time) relational language.

Codd, E. (1985). "Is Your DBMS Really Relational?" and "Does Your DBMS Run By the Rules?" ComputerWorld, October 14 and October 21. Elmasri, R., & Navathe, S. (1994). Fundamentals of Database Systems. 2nd ed. Redwood City, CA: The Benjamin/Cummings Publishing Co. pp. 283 285.