Fermat’s last Theorem $n=4$

Question

I’m revising for my upcoming number theory exam and there is usually an essay style question on it regarding Fermat’s last Theorem, UFD and the method of descent. However, sometimes there have been questions where you’ve had to comment on the difficulties arising from extending $n$ to larger numbers. Could someone maybe just bullet point of the problems that do arise because I (obviously wrongly) thought that the case $n=4$ was easier to prove than the case $n=3$.