forest vs biome

forest and biome both are nouns.

As nouns, biome is a hypernym of forest; that is, biome is a word with a broader meaning than forest:

forest: land that is covered with trees and shrubs
biome: a major biotic community characterized by the dominant forms of plant life and the prevailing climate

Other hypernyms of forest include ground, earth, land, dry land, solid ground, terra firma.

forest (noun)	biome (noun)
the trees and other plants in a large densely wooded area	a major biotic community characterized by the dominant forms of plant life and the prevailing climate
land that is covered with trees and shrubs

forest (verb)	biome (verb)
establish a forest on previously unforested land