I ask this because someone mentioned ‘it isn't a theory, it's a model’ in the comments in an old question of mine: Is the expansion of the universe proof of the big bang?
I guess defining terms is important. If there are different opinions about this, then it would be great if you state that in your answer.
Is the Big Bang a theory, a model, or perhaps both or neither?
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.
My understanding is that the "big bang" (which I am taking to be a synonym for $\Lambda$CDM cosmology) is a model. It sets out a (relatively) simple framework, starting from a set of assumptions, whereby we can explain a set of observational phenomena and do calculations that enable us to predict the results of other measurements.
It is quite a capable model that does represent quite a wide range of phenomena quite well - ranging from the primordial abundances of deuterium and helium, through the cosmic microwave background to the measured redshifts of galaxies as a function of their distance.
Part of the model is that it contains some poorly understood components (inflation, dark matter, dark energy) that have been introduced phenomenologically to solve various problems with preceding versions of the model. That the model has been tinkered with numerous times is actually a good pointer to the fact we are dealing with a model and not a theory. It also has adjustable parameters (the relative fractions of dark matter, energy and normal matter, the photon to baryon ratio) that are not predicted from any first principles but must be tuned to match the data. Tuneable parameters and the introduction of new parameters are also signatures of a model rather than a theory.
A big bang theory would have an explanation for what dark energy and dark matter is, for the photon to baryon ratio, and would incorporate, in a holistic way, the inflationary epoch. It could make predictions about what will happen in the future - for example what will happen to the dark energy component (the big bang model can only make predictions using the parameters we have measured and an assumption about their time evolution).
An analogy would be comparing the "plum pudding" or Bohr models for the atom with the description provided by the theory of quantum mechanics.
Wikipedia has pages on scientific theories and on scientific models, which seem to contain a sensible discussion. There the distinction is made that a theory should offer both explanation and description, whereas a model need only describe a phenomenon. It is on that basis that I would call $\Lambda$CDM cosmology (a.k.a. the big bang model) a model rather than a theory.
A theory is our understanding of WHY. A model is a description of HOW. A theory is an explanation involving a human: a theory has to be understandable by experts, either in full or piece-by-piece. A model, on the contrary, needs no human; it may even be a mathematical, arbitrarily long and cumbersome description processable by a computer. Since the Big Bang contains both the WHY and the HOW parts, you get both a theory and a model. As human logic and mathematics is inadequate to explain the fundamental quantum-physical postulates but very adequate to describe them, the scales lean towards the model side.
However, relax: some folks say the Big Bang Theory is just a sitcom (hence, the real Big Bang should be a model). :-)
