11 And He Himself gave some to be apostles, some prophets, some evangelists, and some pastors and teachers, 12 for the equipping of the saints for the work of ministry, for the edifying of the body of Christ,13 till we all come to the unity of the faith and of the knowledge of the Son of God, to a perfect man, to the measure of the stature of the fullness of Christ; 14 that we should no longer be children, tossed to and fro and carried about with every wind of doctrine, by the trickery of men, in the cunning craftiness of deceitful plotting, 15 but, speaking the truth in love, may grow up in all things into Him who is the head—Christ— 16 from whom the whole body, joined and knit together by what every joint supplies, according to the effective working by which every part does its share, causes growth of the body for the edifying of itself in love.
We see here that an Evangelist plays an integral part in the Local Church Assembly. He, as well as the Pastor, Equips the Saints for the Work of the Ministry, that is Spreading the Gospel, Making Disciples. He also Builds up the Body of Christ by Teaching, Encouraging, Rebuking and giving Admonition when he is given the opportunity to do so. He Plays an integral part in the maturation of the believer/s. He promotes Godly living, as mapped out through the Rest of the Letter. He Encourages people to Love and Share the Love of God through the Gospel of Jesus Christ. In my personal view, the Evangelist may speak to certain areas of the Church that needs to be addressed (out of season) when the Pastor hasn't had the spiritual insight to do so (for whatever reason).
I believe Evangelists' primary calling is to Call the Sinner to repentance and the Saint to a deeper walk with Christ (during a limited time frame).
I believe the Pastor is to "Do the work of an evangelist" (call the sinner to repentance by Preaching the Gospel of Jesus Christ) as well as Feeding the Sheep a steady diet, Week after week, of the Word of God, both milk and meat.
Edited to Add:
Personally, i believe a major difference of an Apostle, Evangelist and Pastor is the Time Spent with the Congregation (for the most part).
Apostles Started and Established Churches, Evangelists visit to add and strengthen, and Pastors Stay Long term to Feed.