We were going to do Anapurna. But there were poitical problems, a little in Kathmandu, but more so in Pokera near Anapurna. So at the last minute we diverted to The Everest Region instead. Flew into Lukla, and walked up past Namche Bazzare, etc, and towards Base Camp region, but instead of going to Base Camp we shot off to the west to climb a different mountain. Got up to roughly 5,000m.

No matter where you go, you'll enjoy it.

Hillary School and the Khumbu region was very nice. Spent some time around there on the way back down.

Anapurna is the more scenic trek(some big ones) but good to see the biggest one too.
Still if you can't do both I would choose Anapurna hands down. Think about purifying your water as a posed to bottled and a pocket full of pens and little gifts for the kids out in the hills goes a long way. Bartering on food prices is generally not allowed as some poor bugger carried the sh!t a long way for very little return. Where ever you go the place has a magical feel and will live with you forever.