"Next door" refers to the closest neighbor, it doesn't require them to be touching. Your example with states is invalid because you are skipping over a state--that's clearly not next door.
However, if you head west from American Samoa you come to Samoa in probably around an hour (Google can't find a route so I'm guessing how a ship would fare), you pass nothing else in the process. I would call that next door.
There is a scale bar in the bottom right hand corner of the map I posted. From landfall to landfall is about 60km; to take an hour to get there, you would need to be making less than half a knot.
On a calm day, you could get from Western to American Samoa in under an hour, rowing a child's inflatable dinghy. A modern cruise ship steaming past the islands could easily do the journey in 20 minutes (although they would likely not be in that much of a hurry).
That's because the two nations are right next door to each other.
Of course, they have radically different economies, despite their geographic proximity.
American Samoa is a very small colonial outpost of the USA; while Western Samoa is a much larger (3x the population) and independent nation, with strong trading and economic ties to other nations in the region, notably New Zealand and Australia. Indeed in 2011, Samoa moved the International Date Line, so that they would be on the same weekday as those major trading partners. So your short trip would actually take around minus 23 hours to complete; If you set off from Samoa at noon on Saturday the 14th, you would arrive in American Samoa before 1pm on Friday the 13th.
So you are both wrong, and you are both right - they are right next door, and they are not directly comparable economies.
smilie.