Jose invested $\$50,\!000$ for $2$ years at an annual interest rate of $4$ percent compounded yearly. Patricia invested $\$50,\!000$ for the same period of time, at the same interest rate, but the interest was compounded quarterly. To the nearest dollar, how much more money did Patricia's investment earn than that of Jose?
After two years, at a four percent annual interest rate, the Jose's investment will have grown to $50000 \cdot 1.04^2 = 54080$.  Patricia has the same annual interest rate, but compounded quarterly, so each quarter (or each period of three months), her investment is compounded at the rate of $4/4 = 1$ percent.  In two years, there are eight quarters, so Patricia's investment will have grown to $50000 \cdot 1.01^8 = 54143$, to the nearest dollar.  The difference is then $54143 - 54080 = \boxed{63}$.