Each individual lily flower only lasts so many days - sometimes between 4 and 7 days. If your potted one was purchased in full bloom with no more buds left to open, it's finished blooming for the season. Your neighbor's plant might have some unopened buds left to go so it's lasting longer. Asiatic lilies will not give you a second bloom.

Don't water it anymore - you can't stop the normal dying back cycle on lilies. When it's really brown, cut the stalk down leaving 4-6 inches above the soil line.

Your lily will go through it's normal cycle, turning brown and dropping leaves, etc. - this is necessary to replenish the bulb for next season.

Although some people have good luck with them in pots for many seasons - you do run the risk of the bulb rotting from staying too wet over the winter. They're happiest in the ground - I think anyway - and in a spot that affords them as much sun as possible.

Lily bulbs don't need a lot of water - the bulb itself is a self contained unit. In a pot or in the ground, they like soil that drains well.