For Germany:
I don't know much about the German Crown Prince, but I don't see any way he can be worse than his father. My best guess would be that defer mostly to the military, and be less likely to be influenced by political operators like Bulow. Germany will still support A-H in the Balkans, against Russia if necessary, which most likely means eventual war.
The interesting question is what kind, if any, of naval expansion Germany decides to undertake without Wilhelm driving for it. I see the Crown Prince as much more of a traditional Prussian army type, not having his father's obsessions regarding a navy. Germany would most likely have better relations with Britain, perhaps enough to keep them out of a war on the side of Russia/France.
For USA:
Lots of stuff happened during TR's first term that might play out much differently with McKinley still in office, like the coal miners strike. McKinley would be inclined to use traditional methods to open the mines, rather than arbitrate between the unions and owners as TR did. In the short run, this probably works and the coal supply is resumed, but could lead to increased unrest and radicalism.
I actually don't see McKinley handling the canal issue much differently than TR, he was certainly no slouch in the imperialism department, if not as strong a true believer as his veep. This is actually one area where I see TR influencing McKinley, or members of his cabinet.
TR as VP would actually be a very interesting thing to see, if anyone had the potential to make the Vice Presidency relevant, it would be him. Foreign policy would be the logical area for him to try and exert influence and get additional publicity for a presidential run in '04 - especially if McKinley is distracted by domestic concerns like labor unrest.
If McKinley takes a very strong pro-business line, which he is likely to do, in '04 the Democrats will likely nominate Bryan instead of a conservative like Parker, which would be one hell of great presidential race.