Why does Jupiter break comets apart? I understand that there are magnificent tidal forces that come with getting too close to Jupiter, but what is causing comets like Shoemaker-Levy to get torn apart?  

I wouldn't expect that the gravitational forces are so strong that spaghetification occurs, is it?
It is.  Tidal disruption occurs when $\rho_{Jup}(R_{Jup}/R)^3$ > $\rho_{comet}$ where R is the separation and $\rho$ is the density.  The density of Jupiter is 1.33 g/$cm^3$ and the comet was estimated to have a density of only 0.5 g/$cm^3$ based on the radius at which it disrupted.  That is, disruption took place at separation R ~ $(1.33/0.5)^{1/3} R_{Jup}$.