Beurre noisette (also known as brown butter, sometimes 'caramelised butter') is whole butter, while ghee is a toasted form clarified butter. (Unless the person using the word "ghee" is for is unaware of the differences so not using its technical meaning).

Beurre noisette - butter is heated until the milk solids in it are toasted. All the water content is usually cooked off during the browning process. (In black butter the heating just continues to a darker colour.)

Clarified butter is any butter where the milk solids (protein) have been removed. - because there are no milk solids it burns less easily (at a higher temperature). It keeps better than whole butter and has lower (or no) water content.

For the specific kind of clarified butter called ghee (an word from India where the form originated), the butter is heated, the milk solids sink, then heating continues until the milk solids are toasted. The milk solids are then separated (and used in other recipes or discarded). The process gives ghee, unlike other forms of clarified butter, a nutty taste and a different colour. The cooking process required also removes all the water content.

Regular (or classical European) clarified butter is heated to a low simmer so the milk solids float to the surface where they are skimmed off off before they change colour. It has no toasted flavours and, given the shorter process needed, usually has a significant proportion of the water still in it.

Both have a nutty flavour but beurre noisette still contains the milk solids. Ghee doesn't (unless the person using the word "ghee" is for is unaware of the differences so not using its technical meaning).

Beurre noisette (also known as brown butter, sometimes 'caramelised butter') is whole butter. It shouldn't have water content but can sometimes still have some water remaining. Ghee is a toasted form of clarified butter (milk solids removed), and never has any water content. It keeps better than whole butter and has a higher smoke point.

Beurre noisette - butter is heated until the milk solids in it are toasted. All the water content is usually cooked off during the browning process. (In black butter the heating just continues to a darker colour.)

Clarified butter is any butter where the milk solids (protein) have been removed. - because there are no milk solids it burns less easily (at a higher temperature). It keeps better than whole butter and has lower (or no) water content.

For the specific kind of clarified butter called ghee (an word from India where the form originated), the butter is heated, the milk solids sink, then heating continues until the milk solids are toasted. The milk solids are then separated (and used in other recipes or discarded). The process gives ghee, unlike the european forms of clarified butter, a nutty taste and a browned colour (depth of colour to taste). The cooking process required also removes all the water content.

Regular (or classical European) clarified butter is heated to a low simmer so the milk solids float to the surface where they are skimmed off off before they change colour. It has no toasted flavours and, given the shorter process needed, usually has a significant proportion of the water still in it.