Harry's father, doesn't actually want him to do anything. Harry's dad is dead. Harry thinks his father is telling him to kill Peter, and Harry's not hesitating, he goes all out and tries to do it.
Now, there is no reason in the movie why Harry shouldn't have the Green Goblin costume. He should have worn it, but we all know he doesn't, which goes against his character and changes him quite a bit from the comics. And this isn't a small villian we're talking about either, Harry was a main baddie for quite a few years, so not having him be GG2 is really like calling the Joker "Mr. Smiley" in the next Batman movie, and giving him a whole new look. There's just no reason for it.
edit: Actually I take that back. There's was no reason for Harry in the previous two movies not to follow in the footsteps of his comic counterpart, but there is a reason for him not wearing the goblin outfit, and that reason is simple. Sam Rami wanted to do it differently. Is that bad? Well, in this case, I think it was. I liked Harry's character better when he became the Goblin, and I think if he had set it up differently, it would have been a cool opportunity to fool the audience a bit and have them think that norman possibly returned from the dead. (they would have had to change the ending of SM2 to do this, but I think it would have been cool).
Really it all comes down to which version you like, and I like the version I grew up with and that has been established for the past 30 years.