We'll lets start with how are they the same...they are both databases.
What's different is their purpose. A traditional application database is optimized to get data from your front-end application to your back-end database. This type of database is designed and optimize for storage and for quick writes to the database. While we all may be using our databases for long time storage and querying/reporting, these application databases are not optimized for that. Data is normalized, so for you to pull information from the database, you have to join tables...costly for a database.
A data warehouse is optimized for querying. Data is not generally stored normalized and you arent as concerned about storage requirements here. What you want is the ability to execute complex SQL queries and be able to retrieve that data quickly and you dont have to worry about impacting your front end applications. Since your tables are de-normalized, you arent impacted by those complex joins. The main purpose of the data warehouse system is for reporting and analysis.
A data warehouse is a specially setup database designed to hold large amounts of data for reporting purposes. While a normal database is optimized for transactional activity (while keeping a small amount of history) a data warehouse will be optimized for large scale reporting.
Data warehouses are designed to help you analyze data.
For example, to learn more about your company's sales data, you can build a warehouse that concentrates on sales. Using this warehouse, you can answer questions like "Who was our best customer for this item last year?" This ability to define a data warehouse by subject matter, sales in this case, makes the data warehouse subject oriented.
Database is an integrated collection of logically related records or files consolidated into a common pool that provides data for one or more multiple uses.
examples of database are:google, Point of sale Inventory system (Till) Warehouse Inventory System, Electronic Telephone Book Airline booking system Hotel Booking System