According to Valmiki Ramayana, Rama was an avatar of Vishnu and he came to earth with the prime purpose of annihilating Ravana, based on the requests of devas and sages who suffered immensely under Ravana’s mighty power.
His very birth was unnatural because he was born by divine blessing that came in the form of a sweet pudding brought by a divine messenger at the end of Putra Kameshti yaga conducted by Dasaratha. Only after drinking the pudding, Dasaratha’s wives could become pregnant and Rama was born as Vishnu’s avatar to Kousalya.
Unlike Krishnavatar of the next yuga where Krishna demonstrated His divinity at every opportune moment, Rama always underplayed his divinity and was content to show just a perfect human face. That was his greatness.
According to Yoga Vashita, Young prince Rama attained Brahma Gnyana, was gripped with extreme vairagya (dispassion) and became disinterested in leading a life of a royal prince. Sage Vashita advised him that one can remain in whatever status one is living already and a Gnyani is never affected by his external worldly activities; there was absolutely no need to relinquish royal life and Rama agreed.
Only because he was a gnyani possessing total mental equanimity, he could accept the life of a forest-dweller at the behest of Kaikeyi without batting an eyelid, despite enjoying all royal comforts. At the same time, in Ramayana, he has been portrayed as a person easily succumbing to emotional depression too (like when he found Sita abducted by Ravana).
On many occasions, Rama indirectly displayed his divinity. Being a divine, he was above dharma and adharma. He displayed this divine aspect when he killed Vali, because he could boldly face criticism of acting in adharmic way in killing Vali by hiding behind a tree.
He showed his mighty power to Samudra Raja, when his prayers to the lord of Seas was not responded initially. Only when he stood up with his Brahmastra with an intent of drying up the seas, the lord of the seas came and surrendered.
During Lanka war, when Rama fell unconscious due to the attack of Nagastra, Lord Vishnu’s vehicle – Garuda came and removed the poisonous effects of the serpant astra. This is another indication of Rama’s divinity and his status as an avatar of Vishnu.
During the final stages of battle with Ravana, Lord Indira sent his chariot along with the charioteer Madhali to assist Rama. It was Madhali who reminded Rama to use the Brahmastra to kill Ravana. Ordinary human beings don’t get such privileges!
Coming to your second part of question: Why can’t we possess similar characteristics like Rama?
The Vedanta says all of us are potentially divine. In Bhagavad Gita, lord Krishna says He is the in-dweller in all. The only problem is that we are totally unaware of our true status, as we are deluded by Maya and ego.
If only, we can get totally rid off maya and discard our ego, we too become God. That’s is precisely Self-realization or God realization.
But still, do we become equal to Rama? Sri Ramakrishna Paramahamsa explains that while as Brahman (God) we may all be fundamentally same, the expression of divine Shakti in each of us is quite different. An ant and an elephant are God in essence alright, but the power of an elephant is not same as an ant!
An Avatara Purusha is one where God’s power gets expressed in a massive scale. Sri Ramakrishna used to say that an Avatar is like a huge steamer that can carry lots of people to cross the sea of samsara. On the other hand, a normal Jivan Mukta (a person who attained God realization through lots of effort and sadhana) will not have so much power to save many others from samsara. He is at the best like a small boat that can fetch a couple of persons across the river.
When Rama’s avatar came to an end, he took the entire populace of Ayodhya along with him to immerse in the Sarayu river and gave Moksha to all of them. Now you know the difference!