In just a few words...
SAX (Simple API for XML): Is a stream-based processor. You only have a tiny part in memory at any time and you "sniff" the XML stream by implementing callback code for events like tagStarted()
etc. It uses almost no memory, but you can't do "DOM" stuff, like use xpath or traverse trees.
DOM (Document Object Model): You load the whole thing into memory - it's a massive memory hog. You can blow memory with even medium sized documents. But you can use xpath and traverse the tree etc.