In both Sanyaas and Vairagya you let go of materialistic stuff and don't remain connected to physical pleasure. Are they the same and or there are any subtle differences between?
According to the sanskritdictionary.com, the word 'vairagaya' means
वैराग्यम् [विरागस्य भावः ष्यञ्] : Absence of worldly desires or passions, indifference to the world, asceticism
Whereas 'Sannyasa' means
संन्यासः - Complete renunciation of the world and its possessions and attachments, abandonment of temporal concerns
So generally 'VairAgya' is taken as something mental and sannyasa is taken generally as both mental and physical. In otherwords, sannyAsa is a Ashrama (One of the four stage of life,) while Vairagya is possible while one is staying in any Ashrama or stage of life. There are also different types if sannyasa like Abhaduta, bahudaka, kutichaka, hamsa and Paramahamsa etc.Each is generally associated with accepting some external sign or 'linga' of sannyasa. Dasanami sannyasa started by Achaya Adi Sankaracharya is associated with renouncing three desies: for child (putra-eshaNA, for wealth (vitta-eshaNA) and for fame (loka-eshaNa).
It is advised:
yad ahareva virajet, tad ahareve pravrajet : take sannyasa on the very day when you acquire vairagya. (Yajnavalkopanishad 1)
Pantanjali in His Yoga-Sutra defines 'yoga' as 'chitta-vritti-nirodha' and lays huge importance on Vairagya :
YS 1.12, "abhyāsa-vairāgyabhyāṁ tannirodhaḥ" meaning chitta-nirodha or Yoga is possible by practice and VairAgya.
Sri Krishna in Gita however defines sannyasa as
He who sacrifices the fuits of karma is a sannyasi. He is a sannyasi who has no hatred or desire (5/3).
On the otherhand,, He also mentions that
mind can be controlled by practice and 'vairagya'(dispassion) (6/35).
So according to Sri Krishna, Vairagya and Sannyasa in real sense are two sides of the same coin. He gives the maximum importance to Vairagya repeatedly and indirectly discourages external sannyasa.