In the particular edition of this play that I read the editors included and essay by Harold Jenkins (not that that name means anything to me) about whether Henry IV is two five act plays or one ten act play. Personally I don't care either way and would really not want to write a major thesis on that particular point, but that is probably because there is so much more with regards to Shakespearian plays, such as the nature of the human condition, and also the nature of political revolt, that I consider that an essay on whether two plays are one or one play is two is probably just a waste of my time. Then again, each to his own, and if this is what interests Jenkins then who am I to criticise him. Anyhow, my position with regards to that question is that it is neither because I actually see it as one forty act play (beginning with Richard II and ending with Richard III) that has been split into eight parts that, in a sense, each can stand on their own as individual plays.
I recently saw this play performed in Sydney by the Bell Shakespeare Company (which is probably the leading Shakespearian theatre group in Australia) and they had performed the two plays as an amalgamation, however since the entire performance was a little under three hours (excluding the twenty minute interlude) there was a number of scenes that had been dropped, and I suspect most of them were from the second play (the rebellion of Northumberland and the Archbishop Scroop was not included, despite the scene where Falstaff examining troops with Justices Swallow and Silence being included). The play itself, as with most Shakespearian performances these days, had been brought into the modern setting with the nobility dressed in suits and the scenes in Eastcheap done as if it were in a modern Australian pub. Falstaff himself did change his style in this play going from being little more than a bum to being a well dressed bum, however that had something to do with his elevation from being a trouble maker to a knight in the second play.
What I didn't notice in the first play but did notice this time was that Falstaff actually claims the credit for killing Hotspur. We know that Hal kills Hotspur, but leaves the scene before anybody can confirm the kill, and Falstaff, who had been playing dead for most of the battle (which is not surprising) then gets up and puts a knife in Hotspur's body and claims the kill. As such, when the King enters the scene, he immediately strips Hal of the kill and awards it to Falstaff.
Now this is actually an important event, especially for those who claim that Hal's return to his wild ways in the second part is inconsistent with the first part where he goes from being a tavern rat to being an honourable battlefield commander. Firstly, Hal is quite bitter at the award for killing Hotspur going to Falstaff on the grounds that he knows that Falstaff is a liar, a cheat, and incredibly lazy (as well as being a coward). In fact, in the second play Falstaff and Hal only encounter each other twice, and where the only change in Falstaff is his title, Hal's attitude has changed dramatically. In fact of both times that Hal and Falstaff meet, the former is rebuking the latter (the first time is where Hal masquarades as a servant boy to listen to what Falstaff says about him when he is not around, and the second is during the coronation parade when Falstaff foolishly expects that Hal will turn England into a thieves' paradise).
In a way the play of Henry IV (in two parts) is not so much about the redemption of a wayward child (though in some aspects it is) but rather about a boy's journey into adulthood. By the second part, Hal has already been redeemed: the prodigal son has returned and he is not going out again. The only reason he returns to Eastcheap is to see if Falstaff himself has changed, but that is not going to happen. Shakespeare is too realistic with his characters, and it is clear that Falstaff is simply too old to be able to break away from a lifetime of bad habits. It is interesting too that even though Falstaff does not appear in Henry V, many of the other companions from Eastcheap do and form a part of the irregular army. Once again, Hal, in the next play, puts on a disguise and goes and mingles with them, but this time he does not reveal himself, he just listens. In Henry V we learn of Falstaff's fate: in Act 2, Scene 1, when Falstaff's page enters and tells his companions advising that he is sick as the king has broken his heart. However, we never actually hear of his fate (and since the fleet was setting sail to France, and since we know that Falstaff is, well, basically a coward, it is not surprising that he would be hiding under his sheets and not wanting to go and fight a real war).