Back in the good old days, we used words like theory, model, hypothesis, law and so on, and their meanings arose from how the words were used, and dictionaries organised these senses - just like other words. Then someone (probably in America) came up with the idea "We don't teach evolution because it's just a theory." And after much shouting, people decided that words like "theory" had to have "scientific" definitions - the dictionaries weren't good enough, and nobody was to say "just a theory".
And since then people have had these odd discussions "Is evolution a theory or hypothesis"... "Is Newtonian gravity a model or a theory or a law".
So what about "Big Bang" - well it's mostly a slogan, initially formed to ridicule the idea of a hot, dense state of the universe. Later adopted as shorthand for this notion. As such it isn't a model. It isn't a (usually mathematical) description of some natural phenomenon that can be used to make predictions. The ΛCDM model of the universe is a model. Models are usually simplified in some way so that one can use them to make predictions. But sometimes when people talk about the "big bang" they are using it as a shorthand for a range of models of cosmology that have an early dense state.
It is a theory, it is an explanation for the observation that matter in the universe seems to be moving away from us in all directions. Sometimes when people talk about the big bang, they are using it as a shorthand in contrast to a theory that matter is constantly created, or theories that the apparent motion is deceptive (etc).
Sometimes "Big Bang" is just used as a shorthand for scientific cosmology, in contrast to that described in the Hindu Yugas, or the Bible, or other religious or mythological texts.
So the answer to this is 1) not very important. 2) Depends on what you understand these words to mean.
But of the two words "theory" seems to fit better.