Well to start the 100 is just a way to convert the percentage value and isn't really necessary as percentages are typically understood to be decimals anyway.
So now we're left with
$$FV = PV\left(1+\frac{r}{k}\right)^{kn}$$
The intuition here is quite simple, you're just considering '$k$' extra periods per year at '$\frac{1}{k}$'th of the rate. for example instead of an interest rate of $10\%$ per annum for $10$ years we can think of it as $5\%$ per 'half year' with twice as many half years ($20$ half years).
So now we're back down to trying to understand:
$$FV = PV\left(1+r\right)^n$$
Which is simply just stating that the future value of an investment (or loan or whatever else you're compounding) is its present value, iteratively multiplied (n times) by some factor $(1+r)$. And this is just the way we increase a value by a percentage. For example if I wanted to increase $100$ by $20\%$ I would say $100\cdot\left(1+20\%\right)$ which is equal to $120$. If I wanted it to grow twice, I would multiply it again so:
$$100\cdot\left(1+20\%\right)\cdot\left(1+20\%\right)$$
or, equivalently: $100\cdot\left(1+20\%\right)^2$
If I wanted to grow it n times, I would say:
$$100\cdot\left(1+20\%\right)\cdot\left(1+20\%\right)\cdot...\cdot\left(1+20\%\right)\qquad\text{'n' times}$$
or, equivalently: $100\cdot\left(1+20\%\right)^n$
Bringing this idea back to the original formula you shared, if I changed the context to compounding $\$100$ at $20\%$ per annum for $n$ years, but I wanted to compound it every $6$ months ($2$ times a year) I would essentially split my $20\%$ rate in half and compound it twice as many times:
$$100\cdot\left(1+\frac{20\%}{2}\right)^{2n}$$
Where here $k$ from the original formula is $2$.
It is important to understand that this is different from taking the yearly rate and raising it to the power of the number of years because of the nature of compound interest. Every time an investment is compounded, it's growing by it's current value as opposed to growing by it's initial value (like in simple interest) So in the scenario above, compounding it twice a year means that it can grow for the first half of the year and then the next half grows from whatever value the investment had at the half way point (which for positive interest rates will be more than the initial value) so it grows faster. The more compounding periods we have, the greater the $FV$ will be if everything else remains constant (to a limit: if you want to learn more about this limit you can research the number $e$).
$\textbf{Important for building intuition}$
If you want to visualise, this hop onto desmos, or any other graphing website/software, and type in: $y=a(1+\frac{r}{x})^{xn}$ into one line, and $y=a(1+r)^n$ on another. You can set values yourself for present value (a), rate (r) and number of periods (n) or you can just add a slider for them and experiment a little bit!
You should be able to see that when $x>1$, i.e. we're compounding more frequently than once per year, the formula with compounding is greater than the formula without.
Just for fun, while you're on Desmos, type in $y=ae^{rn}$ in another line and you'll see that the compounding function approaches $e$ but never surpasses it - that's what I was referencing earlier on when I mentioned a limit.
An annual interest rate of $20\%$ compounded quarterly means what the formula reflects, namely that your investment grows once every quarter at a quarterly rate of $20\%/4=5\%.$ This is not the same thing as an annual interest rate of $20\%$ (no compounding), which means your investment grows once every year at the yearly rate of $20\%$. As you have noted, the former gives an effective annual rate of approximately $21.55\%$ while the latter simply gives an effective annual rate of $20\%.$
Compounding more frequently thus leads to a higher annual return, i.e.
$$P(1+r/n)^{nt}\quad (1)$$
increases in $n$ for $r,t>0$. Intuitively, the exponential effect of more compounding periods outweighs the linear effect of the lower per-period interest rate. As you suggestively note, taking the limit of $(1)$ as $n\to \infty$ gives the continuous compounding formula
$$Pe^{rt}.$$
Best Answer
They are in fact different formulas and will generally yield different answers for the same values of $i$, $n$, $t$, and $P$.
The first formula gives the future or accumulated value of an investment of $P$ for a term of $t$ years, where the compounding occurs $n$ times per year, for a nominal annual rate of interest $i$.
The second formula gives the accumulated value of the same investment for the same term, but in this case, $i$ is an effective annual rate of interest. In this formula, the frequency of compounding is irrelevant, since substituting $(1+r)^n = 1+i$ yields $$F = P(1+i)^t,$$ which shows the accumulated value does not depend on $n$.
What does "nominal" and "effective" mean in this context? An effective rate of interest is one that specifies the amount of change in the investment over a given time period (usually but not always one year), in such a way that it does not matter what the compounding structure looks like within the period. So if I say $6\%$ effective annual interest rate, that means if I have $100$ in principal at the beginning of the term, then at the end of one year, I will have $106$, no matter whether the investment is being compounded quarterly, monthly, daily, or hourly. The commonly-known and used terminology for effective annual interest is "APY," or "Annual Percent Yield."
As convenient as effective interest may be to understand, it naturally leads to some questions: for instance, in many cases, the owner of such an investment might not hold it for an exact number of years. Or in the case of a loan, the borrower may need to repay the loan on a monthly basis. In such situations, it is of interest to know what is the balance on the investment or loan at some intermediate point within the annual period.
So, one way we can also specify how interest is accrued is to state a "nominal" rate, one that takes into account the frequency of compounding. For instance, if I say that a loan has a nominal annual interest rate of $6\%$, compounded monthly, that means that the effective monthly interest rate is $0.06/12 = 0.005 = 0.5\%$ per month, and on a principal of $100$, the loan balance would be $100.5$ after one month. After 12 months, the loan balance (assuming no repayments) would be $100(1.005)^{12} \approx 106.168$.
As you can see, in this case, the nominal annual rate, $6\%$, is slightly lower than the corresponding effective annual rate, which is $6.168\%$. The commonly-known and used terminology for nominal annual interest is "APR," or "Annual Percentage Rate."
Which one do we use? Simply put, you can use either one--typically, whichever is more convenient--as long as you are clear about which interest rate you are specifying. As we saw in the previous example, $6\%$ nominal interest compounded monthly is equal to $6.168\%$ effective annual interest.
In actuarial practice, we use $i$ to denote effective interest, and the symbol $i^{(m)}$ for nominal interest compounded $m$ times per period (usually a year). Therefore, the relationship between these two rates is $$\left(1 + \frac{i^{(m)}}{m} \right)^m = 1 + i.$$