Basin probably arises from baccīnus, ‑um, and to be a derivative of bacca ‘vas aquārium’ - a water vessel. The word dates back to the 13th century. (from OED)

The OED gives

Basin I.1.a. A circular vessel of greater width than depth, with sloping or curving sides, used for holding water and other liquids, especially for washing purposes.

NB: Other than tipping out the water, there was no way of emptying the basin.

A sink, on the other hand was

I.1.a. A pool or pit formed in the ground for collecting waste water, sewage, etc.; a cesspool. Later spec.: a temporary latrine, as dug on an army campsite.

1413–14 In reparacione Infirmarii..emendacione de le synk, 10 d. in J. T. Fowler, Extracts Account Rolls of Abbey of Durham (1898) vol. I. 268 (Middle English Dictionary)

It then became the means of empty waste into a pool or pit:

I.1.b. A conduit, drain, or pipe for carrying away dirty water or sewage; a sewer. Also: an opening specially made for this purpose. Now rare (English regional in later use).

1499 Cynke of a Lawere, mergulus. Promptorium Parvulorum (Pynson) sig. cv/1

However, between these meaning, there was also

II. A basin used for washing.

II.4.a.A fixed basin made of stone, metal, or other material, designed to hold water for washing and having an outflow pipe; esp. (in later use) such a basin with a water supply.

Traditionally associated with the kitchen but also used (esp. now) to denote basins in bathrooms, laundries, etc.

1440 Synke, for water receyvynge, exceptorium. Promptorium Parvulorum (Harley MS. 221) 456

?c1450In the Chaumber over the Parlor..a lede, a synke. in Archaeologia (1869) vol. 42 404 (Middle English Dictionary)

2005 She did not keep a very neat kitchen—dishes were mounded in the sink. New Yorker 9 May 74/1

The essential difference between a basin and a sink is that a sink can be drained via a hole (usually in the bottom) and has a pipe that provides an outlet for the water that is then conveyed to a suitable place.

However, that was the early difference. Currently, we have indoor plumbing and the two words merged and overlapped to a great extent and became confused. That said, even today, a basin is often used for the container (usually plastic) that is placed in the sink.

Basin probably arises from baccīnus, ‑um, and to be a derivative of bacca ‘vas aquārium’ - a water vessel. The word dates back to the 13th century. (from OED)

The OED gives

Basin I.1.a. A circular vessel of greater width than depth, with sloping or curving sides, used for holding water and other liquids, especially for washing purposes.

NB: Other than tipping out the water, there was no way of emptying the basin.

A sink, on the other hand was

I.1.a. A pool or pit formed in the ground for collecting waste water, sewage, etc.; a cesspool. Later spec.: a temporary latrine, as dug on an army campsite.

1413–14 In reparacione Infirmarii..emendacione de le synk, 10 d. in J. T. Fowler, Extracts Account Rolls of Abbey of Durham (1898) vol. I. 268 (Middle English Dictionary)

It then became the means of empty waste into a pool or pit:

I.1.b. A conduit, drain, or pipe for carrying away dirty water or sewage; a sewer. Also: an opening specially made for this purpose. Now rare (English regional in later use).

1499 Cynke of a Lawere, mergulus. Promptorium Parvulorum (Pynson) sig. cv/1

However, between these meaning, there was also

II. A basin used for washing.

II.4.a.A fixed basin made of stone, metal, or other material, designed to hold water for washing and having an outflow pipe; esp. (in later use) such a basin with a water supply.

Traditionally associated with the kitchen but also used (esp. now) to denote basins in bathrooms, laundries, etc.

1440 Synke, for water receyvynge, exceptorium. Promptorium Parvulorum (Harley MS. 221) 456

?c1450In the Chaumber over the Parlor..a lede, a synke. in Archaeologia (1869) vol. 42 404 (Middle English Dictionary)

2005 She did not keep a very neat kitchen—dishes were mounded in the sink. New Yorker 9 May 74/1

The essential difference between a basin and a sink is that a sink can be drained via a hole (usually in the bottom) and has a pipe that provides an outlet for the water that is then conveyed to a suitable place.

However, that was the early difference. Currently, we have indoor plumbing and the two words merged and overlapped to a great extent and became confused. That said, even today, a basin is often used for the container (usually plastic) that is placed in the sink.