I've been to both. I used to travel to Hawaii and Guam often for work. I had the wife meet me there on one occasion and it was great. I worked on Oahu. We did our first few days touring that island. If you plan on visiting that one, stay away from Honolulu! It's crowded and has a f'ck ton of traffic. The Ko'Olina Beach resort next to the Disney resort on the west end of the island is awesome. Private man-made lagoon style beaches (no massive waves, easy swimming) are great. The enormous pool, grotto/waterfalls/swim-up bar is badass too and you can watch sunsets since it's on the west side.
Then head up north to visit the Dole plantation....best pineapple i've ever had! Continue north to the famous Oahu north shore to some great beaches (Waimea) and good snorkeling sites. On the north shore and northeast side there are some great roadside shrimp trucks. The scampi is out of this world and stupid fresh!
After that we flew over to the Big Island and stayed one night in Hilo. It's wet and tropical and close to the Volcano National Park. Definitely a must-see! Drive down to the coast where the lava fields drop into the ocean and walk around on it. Pretty cool stuff. You can hike all the way to where the active lava is pouring into the ocean, but when we were there it was a three-mile hike! We then drove the south loop to the southern most point in the US, and continued up to Kona for a few days. If you like coffee, visit the plantations...our favorite was Kona Joe's. Kona has great beaches for swimming/snorkeling. There's Turtle Beach nearby, aptly named as I saw plenty of them while snorkeling there. We also took a dolphin swim charter out from there which was fun.
Talk to the locals and they'll give you the scoop on all the good beaches. We were told about one that was through a gated private community. It was nice black sand and hardly any people. In Hawai'i they are not allowed to make access to the ocean private, so even though it's gated, they still have to let you in. Just tell the guard shack you're visiting the beach. If you like mountains, you can also drive to the top of Mauna Loa to the observatory at 14k feet or so.
We went in November around Thanksgiving and it was plenty warm enough for swimming, less tourists etc. My next visit I want to hit Maui and Kauai. I hear they're amazing and the Na'Pali coast on Kauai is badass and pretty world famous.
Now, as far as American Samoa....fuck that place. It sucks ass. I went there for work too and it's hot, humid and I've never seen so much wanton litter in my life! Every inch of every ditch on that island is filled with garbage. It's a pandemic and is everywhere! Also, unlike Hawai'i, every inch of that island is privately owned with no beaches or public beach access unless you're a Samoan and live there. Nice people, but damn, what a shame. They had a billboard campaign across the island trying to get them to stop littering but it didn't seem to help. This was 2009 so maybe it's gotten better? Who knows?
Hope that all wasn't too redundant since you were there for a month. Let me know if you have any other questions Devil Dog.