Honda doesn't make a car the competes equally with an Avalon, the closest is the Accord. I would say the reliability is the same, but Toyota has a superior fit and finish and a more refined drive.Pound for pound honda's have a bit more performance speedwise.
Toyota and Honda are both very good car companies. They have very similar cars except Toyota builds generally larger cars.
Hondas are known for great crash test ratings but Toyota is still up there.
I would pick a Toyota but it all depends on your values.