The main difference from a user perspective – which I think the previous answer does not stress enough – is that Metaspace by default auto increases its size (up to what the underlying OS provides), while PermGen always has a fixed maximum size. You can set a fixed maximum for Metaspace with JVM parameters, but you cannot make PermGen auto-increase.
To a large degree it is just a change of name. Back when PermGen was introduced, there was no Java EE or dynamic class(un)loading, so once a class was loaded it was stuck in memory until the JVM shut down – thus Permanent Generation. Nowadays classes may be loaded and unloaded during the lifespan of the JVM, so Metaspace makes more sense for the area where the metadata is kept.
Both of them contain the java.lang.Class instances and both of them suffer from ClassLoader leaks. Only difference is that with Metaspace default settings, it takes longer until you notice the symptoms (since it auto increases as much as it can), i.e. you just push the problem further away without solving it. OTOH I imagine the effect of running out of OS memory can be more severe than just running out of JVM PermGen, so I’m not sure it is much of an improvement.
Whether you’re using a JVM with PermGen or with Metaspace, if you are doing dynamic class unloading, you should to take measures against classloader leaks, for example by using my ClassLoader Leak Prevention library.