What is the difference between mutation and variant? News reports last week called Omicron a mutation of SARS-CoV-2; but this week they are calling it a variant and they are adding that it has a lot of mutations. So I am puzzled.
A mutation is just a genetic change. Mutations include point changes that are simply changes of one base to another (e.g., an "A" to a C, G, or T/U), insertions or deletions of single bases, or larger changes like insertion or duplication of a whole chunk of DNA/RNA.
Variants for viruses are a bit like a "subspecies" or perhaps even species for other organisms. Species definitions more generally are somewhat arbitrary - they are useful categories, and one can define criteria to identify species, but no one criterion is "more correct" than another, and every set of criteria we have come up with so far is dissatisfying in some fashion, such that it isn't typically followed strictly.
The important consistent thing about species names across all biology, though, is that they are labels for a lineage, a group of entities related by descent.
I've written an answer about virus variants on Biology.SE. Quoting from my own answer there:
A virus "species" is just a label to a lineage at one point in time. When we discover separate lineages of a virus, historically they have gotten a new label. It's then possible follow the lineages from there into strains/substrains, etc.
It would be possible to give a new lineage name to each and every virus particle that has a mutation that makes it different from the parent virus it was copied from, but this wouldn't be very practical. Most of these mutations or combinations of mutations aren't practically relevant, so even if these are all technically variants it's not necessary to give each a name (instead you'd refer to the location and substance of the mutation(s)). If they either don't become prominent in the population of viruses of a given type, or if they behave exactly like their parent viruses, there isn't much use in giving a new name. However, when a mutation or group of mutations (and, practically, it's usually going to be a group, even if many of those mutations are not influential) cause part of a virus lineage to stand out in some way (different infection behavior, higher transmission/prevalence) it becomes useful to give it a name, and that's what the named "variants" are that you hear about in the news: names given to lineages that have diverged from the parent virus.
There are study groups that sit around and talk about these things to decide when we should give a new name and they have some criteria they use for guidance, but ultimately the purpose of these names is entirely practical. See also https://www.who.int/en/activities/tracking-SARS-CoV-2-variants/
In the popular press, you'll probably find that people use terminology a lot more loosely than they do in scientific venues. A variant is a virus lineage that shows one or more impactful mutations from an ancestral version of that virus, so you might hear someone call it a "mutant strain/line/etc".