The only difference between the two is the level of normalization of your dimension table(s); think of star schema as of denormalized snowflake schema.
Here is an example:
Employee dimension is linked to Employee_Earnings fact table by Empl_id.
Let's say each Employee has atributes like Dept, Manager, Division, Region. You can decide to list all these attributes in each Employee record in the Employee Dimension table - than you'll get many repetitive data in this table. And this is perfectly fine with Star schema concept.
Someone else may decide to separate repetitive Dept, Division, Region info into a second dimension table called Dept and provide unique keys for all combinations. Then he would replace Dept info in the Employee table with a reference to the Dept table by Dept keys. Thus, he created two dimension tables linked by Dept key while only Employee is linked to the fact table. He just normilized a Star schema into a Snowflake schema.
Clearly, Snowflake schema is a step back towards to highly normalized OLTP architecture, and away from DW OLAP denormalized architecture.
Hope this helps.