In Part 1, Falstaff was a father-figure to the wild prince and showed him the ins and outs of leading the life of a common thug. As much fun as that was (for us and the prince), once Hal becomes king, he needs a new mentor. (See above.) Enter the Lord Chief Justice, who spends every minute of his time upholding the law, even if it means angering the heir apparent (that would be Prince Hal).
Falstaff, on the other hand, spends all of his time carousing and getting into trouble – whether it's swindling Mistress Quickly out of cash or taking bribes from soldiers, Falstaff's life is all about riot and disorder. Interestingly, when Falstaff learns that Hal has been crowned king, he gloats "the laws of England are at my commandment" (5.3.15). He's wrong about this of course. Hal knows that Falstaff is too corrupt to remain in the king's life. Sadly, this is exactly why Falstaff has got to go. If you're interested in Hal's rejection of his old pal, check out our "Character Analysis" of Falstaff.