burden vs weight down

burden is a noun but weight down is not a noun.

Word	Noun	Adjective	Verb	Adverb
burden	Yes	No	Yes	No
weight down	No	No	Yes	No

As verbs, burden and weight down are synonyms defined as:

Other synonyms of burden include burthen, weight.

burden (noun)	weight down (noun)
weight to be borne or conveyed
the central idea that is expanded in a document or discourse
an onerous or difficult concern
the central meaning or theme of a speech or literary work

burden (verb)	weight down (verb)
impose a task upon, assign a responsibility to	weight down with a load
weight down with a load