Is wanting to do something = desire ?
Not really. Wanting to do something is a thought process.
In Buddhism, desire (Taṇhā) refers to greed or craving pleasure, material goods, and immortality, all of which are wants that can never be satisfied as the world is impermanent in nature and it is always subjected to change. This causes suffering. People generally desire a better life, family, children, money, cars, boats, planes, vacations etc. All these things which are impermanent.
Enlightened people do not possess material craving. Followers of Buddhism tries to mitigate these desires and understand that everything is impermanent. Yes, they do feel hunger, they do feel pain and pleasure but they understand that both pleasures and sufferings are impermanent. For example, a normal person might crave luxurious food while an enlightened person might only eat food just to survive and stay alive.
"Monks have intention to teach dhamma to others"
The monks intention to teach Dhamma is not done for the pleasure or for greed of spreading the religion. If I am not mistaken, I think there are some guidelines for monks when leading an ascetic life and some of them do require a monk to teach so more people open their eyes.
How a enlightened monk differs from an enlightened lay person ?
A monk would abide by the guides related to being a monk such as preaching Dhamma and any other responsibilities set around a monkhood. Lay person is not given these rules hence might live a quiet life. Either way both have understood the impermanence taught in Buddhism and will have no material craving. If by enlightment you mean attained nirvana, they will no longer have another life. ( Remember there are stages of enlightenment. More explained here.