Hi Andrea,
They are "next" to one another. It is walkable. From the ticket booth Neuschwanstein is one direction and Hohenschwangau is in the other direction. There is a bus that takes you up to Mary's Bridge and from there you walk downhill to Neuschwanstein. There is also a horse and carriage that will take you up to Neuschwanstein but leave you below it with a bit of an uphill walk.
We took the bus up, walked along the bridge, walked down to the castle, took the tour, walked down and took the carriage back to the ticket booth/parking area.
Hohenschwangau is much easier to walk to than Neuschwanstein. Not nearly as steep. You can also take a horse and carriage to Hohenschwangau.
If you have time, there's a very good luge ride (sommerrodelbahn) and cable car about 1 mile from the castles.
www.tegelbergbahn.de/227.0.html
www.tegelbergbahn.de/208.0.html
Paul