The actual F-Statistic I come up with between the two tests is pretty similar.
I don't know that this is generally the case. If the population variances are quite different, they should both tend to be large together, but they don't respond to things in quite the same way,
Why is the critical F-Stat for Levene's test so much larger?
Simply because one of the two degrees of freedom is very small. If either df is small, the tail quantiles are larger.
Am I calculating the critical values for the tests correctly?
Well, the degrees of freedom are right -- if you want an approximate 5% test in both cases, then no.
The Levene test only uses the upper tail (that is, it doesn't matter which group has the large variance, it's always an upper tail test). The appropriate upper tail critical value for the Levene test is actually 4.3248 (but it's only approximately a 5% test)
[The variance ratio test is also only approximate, because the "taking the maximum ratio" part in effect treats the degrees of freedom as interchangeable. It's possible to avoid that effect, though.]
Of the two, the Levene test is much more robust to departures from normality.
However, if you're doing this in order to check the assumptions of some other procedure, such as an ANOVA or t-test, the advice of quite a few papers is to avoid such formal testing (particularly if you'll choose a different test when you reject than if you don't).