burst forth vs rush out

burst forth and rush out both are verbs.

Word	Noun	Adjective	Verb	Adverb
burst forth	No	No	Yes	No
rush out	No	No	Yes	No

As verbs, burst forth and rush out are synonyms defined as:

burst forth and rush out: jump out from a hiding place and surprise (someone)

Other synonyms of burst forth include leap out, sally out.

burst forth (verb)	rush out (verb)
be unleashed; emerge with violence or noise	jump out from a hiding place and surprise (someone)
come into or as if into flower
jump out from a hiding place and surprise (someone)