How is rosary different from chanting Mantra? I'd like to know the Catholic teaching or opinion on rosary similarities with eastern religion prayers like chanting mantra.

I checked something about chanting mantra and it uses the same way of focusing on every word and even the way how it sounds.
According to Merriam-Webster, a mantra is 


  a sound, word, or phrase that is repeated by someone who is praying or meditating


So I suppose that one might consider a Hail Mary or the other prayers used in the Rosary as a sort of mantra. Whether one would really want to depends on your view of meditation and meditative prayer.

Merriam-Webster's Concise Encyclopedia entry for mantra states:


  Repetition of a mantra can induce a trancelike state and can lead the participant to a higher level of spiritual awareness.


That's not the Catholic intention when engaged in meditative prayer. The US Conference of Catholic Bishops' document on How to Pray the Rosary states:


  The repetition in the Rosary is meant to lead one into restful and contemplative prayer related to each Mystery. The gentle repetition of the words helps us to enter into the silence of our hearts, where Christ's spirit dwells.


The Catechism of the Catholic Church further adds in paragraph 2723:


  Meditation is a prayerful quest engaging thought, imagination, emotion, and desire. Its goal is to make our own in faith the subject considered, by confronting it with the reality of our own life.


And in paragraph 2705:


  The mind seeks to understand the why and how of the Christian life, in order to adhere and respond to what the Lord is asking.


In other words, this meditative prayer is not intended to lead one into a trance or semi-hypnotic state, though I suppose one could argue that it's intended to produce a "higher level of spiritual awareness". Instead, it's intended to help one focus on a spiritual saying, event, or person, and allow God to show us how we can apply the spiritual meaning contained therein to our own lives.

In a loose sense, then, one could say that the Hail Mary as used in the Rosary is a sort of mantra, but it's not intended to be used for the same purpose that a mantra such as om mani padme om is. There are similarities, and again differences.