Normal Filter:- This is used to restrict the data from database based on selected dimension or measure. If you drag a dimension into filter shelf you can filter the data by selecting list of values. when dragging by measure you can select a range of values.
Quick Filters:- what ever you have applied filter, if you want give a chance to user for dynamically changing data members at run time this is useful. In this you are able to see only non restricted data members only.
The quick filter is a representation of filter to play around and normal filter will not be shown in the workbook to play around.
Filters narrow down the data and there are multiple types (i.e. extract filters, datasource filters, local filters). Quick filters are just a visual representation of the local filters. At least this is how I think about it.