"the problem is that those "breaks" are not stored. "
The breaks are stored, but they are stored as normal ASCII linebreaks, and HTML does not render those. HTML only renders BR tags as linebreaks.
Use the nl2br() function to convert the newlines to BR tags when you print the data on the page
Do not use nl2br() on the data before you put it into the database, because that makes it very difficult to edit the text again in a textarea. nl2br() generally replaces the newline by a BR tag followed again by a newline, so repeatedly calling nl2br() on the same string would add more and more BR tags...