To compare the difference between King Henry and Prince Hal's style of statecraft, first we have to understand the basic philosophy of each. The King belives that to effectively lead the country one needs to lead by example.
According to the King's philosophy the best man is the one who lives a pure life and garners respect and honor from all men. To the King's way of thinking
Hotspur is more fit to be a King than Prince Hal, a comparison the King makes several times. In Act I, scene i King Henry makes his first comparison of
Hotspur to his son saying that Lord Northumberland's son, Hotspur, was "A son who is the theme of honour's tongue
..." while Prince Hal was stained by "...riot and dishonor...." In fact the King goes so far as to wish that Hotspur was his his son and not Prince Henry. Later in Act III, scene ii King Henry tells the
Prince that Hal reminds him of the way King Richard acted before Henry took the throne and that Hotspur reminds the King of himself. This is the King's not so subtle way oftelling Hal that the King doesn't think he is fit to succeed him to the throne.
Prince Hal on the other hand has a different idea of statecraft. He expresses his ideas in his speech in Act I, scene ii when he says "If all the year were playing holidays, To sport would be as tedious as to work; But when they seldom come, they wish'd-for come,...My reformation glitt'ring o'er my fault, Shall show more goodly and attract more eyes than that which hath no foil to set it off." He is saying that he may be acting like a common ruffian now but when he does take up his duties as heir to the throne and begin to act like a prince he will shine more in the eyes of men because his actions now will provide a contrast by which they can judge him. It is important to note that Hal does not plan to spend the rest of his life acting in this manner. He does plan to leave this life behind and take on the role of heir apparent