What is the difference between a golden triangle and a golden gnomon?

The golden triangle is uniquely identified as the only triangle to have its three angles in 2:2:1 proportion.

The golden gnomon is uniquely identified as the triangle having its three angles in 1:1:3 proportion.

A golden triangle is an isosceles triangle in which the two longer sides have equal lengths and in which the ratio of this length to that of the third, smaller side is the golden ratio:

Golden triangles are found in the nets of several stellations of dodecahedrons and icosahedrons. Also, it is the shape of the triangles found in the points of pentagrams.

Closely related to the golden triangle is the golden gnomon, the obtuse isosceles triangle in which the ratio of the length of the equal (shorter) sides to the length of the third side is the reciprocal of the golden ratio. The golden gnomon is also uniquely identified as the triangle having its three angles in 1:1:3 proportion. The acute angle is 36 degrees, the same as the smaller angle of the golden triangle.

https://en.wikipedia.org/wiki/Golden_triangle_(mathematics)

The golden gnomon is the obtuse isosceles triangle whose ratio of side to base lengths is given by 1/phi=phi-1, where phi is the golden ratio. Such a triangle has angles of 36 degrees-36 degrees-108 degrees and can be constructed from a regular pentagon as illustrated above in red. The corresponding 36-72-72 triangle with side-to-base ratio phi is a golden triangle.

http://mathworld.wolfram.com/GoldenGnomon.html