The muakkadah Sunnah is the Sunnah of  which the Messenger of Allah (peace and blessings of Allah be upon him) always performed, and he never left it without any serious excuse, while the non-muakkadah Sunnah is the Sunnah of the which the Messenger of Allah (peace and blessings of Allah be upon him) performed,And sometimes performed it and sometimes left it without any excuse.

The Holy Prophet (peace and blessings of Allah be upon him) himself  emphasized to performed twelve rak'ahs of Sunnah during the day and night. Those twelve rak'ahs are: 

Two rak'ats before Fajr, four rak'ats before Dhuhr and two rak'ats after Dhuhr, two rak'ats after Maghrib and two rak'ats after Isha.

  The difference between  Muakkadah and non-muakkadah is that performance of Sunnah muakkadah is obligatory and whoever skips this prayer without any excuse is a sinner, while performing the non-Muakkadah Sunnah gets a lot of reward but there is no sin in skipping it without any excuse.  Since the Sunnah is in the status of unconfirmed nafl prayer, only the rulings of nafl prayer are issued on it.

The method of performance of muakkadah And non-Muqedah Sunnah is the same, as  there is no significant difference between the two, it is wajib to recite Surah Fatiha in each rak'ah , and three short verses or one long verse along with it.  However, the only difference is that in the four rak'ahs of the non-mutual Sunnah, every two rak'ahs is a permanent prayer;  Therefore, it is better to recite Durood Sharif and Dua in the Qaada of the second rakat, then to stand up for the third rakat, and after standing for the third rakat, recite Sana and Tawz before Surah Al-Fatihah.  Tasmiyyah should also be recited, but this method is preferable, it is not necessary, so if someone recites the non-Muqedah Sunnah and recites only Tahiyat in the first Qadah like Sunnah Muqedah and does not recite Sanaa in the third rakat, his prayer will be valid.  While it is necessary to stand for the third rak'at immediately after reciting tashahhud in the first qa'dah of the four-rak'at sunnah muqedah, in case of reciting Durood Sharif by mistake or omission, sajdah sahu will be required due to the delay in standing.  Similarly, in Sunnah Muqidah, Sanaa is recited only in the first Rak'at.

Another difference is that since each intercession (two rak'ats) of Sunnah non-Muqedah is permanently separate and the Qada of each intercession is counted as the last Qa'dah, therefore, in the four-rak'at Sunnah non-Muqedah prayer, if in the first intercession,  If you don't sit in Qa'da (for reciting tashahhud), that intercession will be invalid, however, if you do not sit in the Qa'da of the first Qa'da, but do Sajdah in the last Qa'da, then the prayer will be valid.