Just like humans are emotional they are also logical. If one cannot make someone feel good about something and also make it sound logical to a person they won't believe it. Whether they are "enlightened" or not.
Now I am not encouraging agruing or debating matters with people because at that point you can see they just aren't ready to be guided by you or Allah. But discussion is good, even if it just means sharing information.
I walked up to an athetist yesterday while I was handing out Islamic material at a festival and she attacked me verbally with sacrasm. So I patiently answered her questions. I found sharing what we had in common instead of our differences really helped. And I used analogies to help her picture things in her mind. By the end of the conversation she walked away with a pamphlet and she had even introduced me to her husband and left with a smile.
This was an example of a successful Dawah attempt. Maybe she will or will not become Muslim one day but maybe Inshallah she will be tolerate of the next Muslim she meets and she understands a little more.
But I find often Muslims boggle people down with speechs and too much information. I was watched some brothers giving Dawah yesterday and they sounded like they were giving a college lecture on the "oneness of God". I thought to myself are those people really going to remember all that information and does it emotionally move them at all.
I find the greatest tool to "evangelism" is personal stories. When people see how Allah has changed your life then they begin to think and they remember that conversation. They might even mention the conversation to someone else and it might start another conversation on Islam. The point is to get the conversation going. Because one thing I have learned as a Muslim is the more knowledge you learn and the more "conversations" you have on Islam the more it attracts you. That's why I believe after 911 so many converted because it started the conversation. And now Ahamdulillah the conversation hasn't stopped.
So as a Muslim even if you don't think you are the "best Muslim" or have the most knowledge think of examples of how Allah has affected you or how the Prophet's actions have changed your life. And share that.