By clicking "Flush Magento Cache" Magento will delete Magento cache which was generated by Magento. Take into consideration that cache storage (for example memcached) can store information not only for Magento. So, that action will not affect anything except Magento's cache itself.
"Flush Cache Storage" will make hard clean up for the whole cache storages at all (like "flush_all" in memcached or "rm -rf ./*" for files).
Normally, we recommend to use "Flush Magento Cache" only - to avoid other data to be flushed.