The Statute of Tenures of 1660 was to do with Crown tenures - i.e. those tenancies created by the Monarch. These would be people of relatively high rank in the feudal hierarchy.
The Act would have had no direct effect whatsoever on the tenancy of the "average peasant Joe".
The Act abolished tenens in capite, and tenancy by knights-service. This meant that, from 1660, crown tenants no longer owed military and religious service to the Monarch. Instead they were required to pay a clearly defined, fixed rent payment ("socage"), governed by Common Law.
Most sub-tenants of those crown tenants often already held their land by socage by 1660, although some other forms of tenancy, such as Copyhold, remained in force. These arrangements remained unchanged by the 1660 Act.
The change in the type of tenancy also had important implications for inheritance. Prior to 1660, lands temporarily reverted to the demesne of the crown on the death of the tenant. If the heir was underage, they became a ward of the crown under the jurisdiction of the Court of Wards and Liveries. That court was also formally abolished by the 1660 Act (although it had been effectively abolished in practice when the Long Parliament abolished feudal tenure in 1646).
The revenue from that court was an important source of income for the crown. To compensate the crown for that loss, a new, annual, fixed payment of £100,000 was enacted. This payment was to be raised by means of a new tax on alcohol.
Additional Info on Law of Real Property
Following the Norman Conquest in 1066, William I imposed a feudal structure on England. He was the owner of all land in England by conquest. He granted land to certain of his subjects in return for services and those subjects might in their turn grant that land, or parts of it, to others, again in return for services. Via land tenures, lower lord owed services to his lord and so on up the pyramid to the Crown. The Statute Quia Emptores (see comments) was used to control this and is the start of modern land law (still in effect).
Before the Tenures Abolition Act 1660, the consideration for getting land was more than just payment/money, i.e. "in return for services" normally meant providing knightly services, military equipment, etc. (tenures of chivalry). There were also two socage tenures: petty sergeanty and common socage. Petty sergeanty involved services of a non-personal nature such as supplying the lord with arrows, or straw for his bed, while common socage usually required services of an agricultural nature, such as ploughing for the lord for 30 days a year.
The Tenures Abolition Act did away with the various tenures (of feudal control), and kept essentially common socage. Megary and Wade, The Law of Real Property (Sweet & Maxwell, 2012) - emphasis mine, p.30:
The system of landholding in return for services fell in decay ... The Crown assiduously preserved the incidents of military tenure, such as wardships, for the sake of revenue. However, following the restoration of the monarchy, the Tenures Abolition Act 1660 converted all tenures into free and common socage ...
Economic historians would call this creating a market (i.e. one can now freely buy/sell land, which was not quite possible previously because of existing conditions, i.e. feudal ties/tenure). In this way, you could say the Act reduced feudal ties via land tenure.