The \$1699.24 number you calculated is the amount paid if you never pay anything against the principal of the loan until the very end. However, in each payment, you are reducing the principal of the loan, such that for each subsequent payment you are paying less interest. Note how this is a different situation to leaving money at a bank without taking any out, thus reducing the principal of your loan to the bank.
Where $p$ is the monthly payment, $i=0.666\%$ is the monthly interest rate, and $V_x$ is the amount owed after the $x^{\rm th}$ payment, this loan is described by the equation
$$V_{x+1}=(1+i)V_x-p$$
This sort of equation can be solved using z-transforms, and perhaps a few other ways, but for this application it suffices to confirm that the given value of $p$ indeed gives $V_{24}=0$.
Using a computer to evaluate the above equation to find $V_{24}$, we get within \$0.20 of zero, confirming that the given price per month is likely correct to within \$0.01.
To contrast, consider the equation where you make no monthly payments. We have
$$V_{x+1}=(1+i)V_x$$
which is easily reduced to
$$V_x=(1+i)^xV_0\rm.$$
You may recognize this formula as the one you used for compound interest.
Using the z-transform as mentioned above, the equation with monthly payments can be reduced to
$$V_x=V_0(1+i)^x-\frac{p}{i}\left((1+i)^x-1\right)$$
which can be solved for $p$ to find the monthly payment.
