There exists a widespread separation of
one year into 12 roughly equal slices. What is the benefit of introducing a very similar one
b = floor(a/4)? How is
we'll do it 2021Q1 better than
we'll be done by Apr 2021? Why do whole organizations plan in terms of quarters?
People have been using fractions of an year for ages to evaluate things and measure progress. Business and project management isn't any different. You measure per year, semester, trimester, quarter, monthly, or by any other interval you consider relevant for whatever you want to evaluate or for gaining information that allows you to take decisions on what to do next.
You won't find an exact reason or a specific cause for why you have fiscal quarters. There is this quote attributed to Gerald Weinberg that perfectly summarizes it:
Things are the way they are because they got that way
It's the law, it's for accounting purposes, for building budgets, for tax payments, for reporting, for investors to think they can make some money fast in a rapidly growing business (see, for example, Has quarterly capitalism made us overhasty? and The Pitfalls of Quarterly Capitalism), etc.
How is we'll do it 2021Q1 better than we'll be done by Apr 2021?
It isn't necessarily better, but it is more familiar. You have a common language with others in the company to align goals and objectives. You could consider Apr 2021 as a milestone, and give that milestone a name like "Milestone 1", which isn't any different than calling it "Q1" if you already know what that means and what other things (like reporting, taxation, etc) need to align with it.
EDIT: to address the comment of "unless I meet a person who knows what Q1 is but has never heard of April".
I think you are ignoring the context in which fiscal quarters - as you say - are a thing. This isn't something everyone uses in their day to day life (like everyone knowing what April is). This is something used in a business, regulatory and financial contexts. People working in this kind of business know very well what Q1 is, just like they very well know what April means, but Q1 has more meaning than April for them. Again, as I mentioned above, that's just how things evolved and turned out to be. April could have been a milestone for reporting, taxation, planning, etc, just as Q1 is, but it just happens people use different terminology.
Here are some other examples from a financial context: we say "bull market" and "bear market". Why don't we say the market is rising, or the market is declining? Everyone understands what rising and declining mean, and what bulls and bears are (animals right?), but not everyone understands what "bull market" and "bear market" means when they hear them mentioned. That's just terminology used in the context of stock markets. How exactly it turned out that way I don't know, but that's what it is. Things are the way they are because they got that way.
In another comment you mention, "Finance is important, but so is sweeping the floor or restocking coffee. Most employees have nothing to do with finance, floor or coffee." That is correct, but that's not what a business is. For example, people sweeping the floor in a bank are not in the business of banking. They do not plan budgets, they do not plan projects and investments, they do not plan goals for Q2 based on what the results of the financial institution were in Q1, they do not try to maximize profits to make the company more attractive to investors. Ask a janitor what Q1 is, and they won't know. But ask any business persons in the same organization what the cheapest brand of floor detergent is, and they won't know either.
When I said "it is more familiar" above, I meant within the business context where everyone uses Q1, not in every day life or for people in the organization that have nothing to do with how the organization plans and measures their business.
This financial analysis divided by periods of the year (quarters, for example), is an important practice to analyze the performance of companies and to be able to exemplify the success or failure of their launches by means of concrete numbers, since these results are transparent, official and required by law. This is also a good way for its shareholders and partners to closely monitor the growth and performance of their brands.
In some countries the fiscal year can vary a lot from the normal calendar (reaching almost six months of difference with the normal calendar). Microsoft is an example of this: its fiscal year starts on July 1st and ends on June 30th the following year.
I understand that for a consumer ("normal person") it'd be much easier to understand
we'll be done by Apr 2021. Yet, the other notation
we'll do it 2021Q1 is better for standardization (imagine having to explain all the time what Q1 means in all of the different languages). Still, as you may know, while these reports are publicly available, they are often explained by analysts so that consumers understand the meaning of the results presented. In that explanation the costumer gets already something more understandable.
What I don't understand is why whole organizations plan in terms of quarters
Planning in quarters isn's something only done by organizations. One could ask it similarly «Why pregnancy is divided in quarters?» and the answer would be that the division by quarters helps the mother to understand the main changes that her body will go through and, in a way, also contributes to accompany the baby's development.
This does't exclude the necessity of a strategic plan of at least 3-5 years (which would match what the mother looks for in 9 months time - her end goal is to give birth to an healthy baby). However, only that wouldn't be enough as it is very easy to lose focus on what you need to do now to achieve these business goals in the long run. So, having a quarterly business plan is critical to growing one's business.
For my work, quarters are a useful artifact. I do long term scheduling, and I need a scope of work that is longer than a month but shorter than a year. I'm looking ahead between 0 and 4 years and trying to schedule work. My customers are comfortable picking a fiscal quarter, but probably would be overwhelmed if they tried to pick a month. Fiscal quarters also allows them to deconflict the schedule against other processes that have a similar scope of planning (e.g. don't schedule lines of business in the last quarter of the fiscal year because they're doing end of year processing)
In summary, mine is one of a number of processes where 90 days/3 months/1 quarter is the appropriate timespan.