What is the difference between an Apostle and an Evangelist according to Charismatic Christianity?

Rather simple, if one sticks to the documented will of JC, and not all the later attempts to get in on the action.

Twelve disciples were directly selected by JC in his travels through Galilee. The disciples are enumerated in Mark and Luke and include Judas.

About two years before the crucifixion, and shortly before the Sermon on the Mount they were designated by JC as his messengers [literally "sent away"], apostles in Greek.

Four Gospels are in the Canon since the late 2nd century, and their authors are the original Evangelists. BTW, this makes Matthew the only Apostle Evangelist.

I will answer both questions that are asked. The first question is whether the perceived biblical role of apostles is the same as that of evangelists.

Apostle - Friberg Lexicon 3299 (1) as one who is sent on a mission with full authority apostle, messenger, envoy (JN 13.16); (2) as a commissioned representative of a congregation delegate, missionary, representative (2C 8.23); (3) in the NT used especially of a messenger for God; (a) generally (LU 11.49); (b) more specifically as a person who tells the gospel message apostle (RO 16.7); often of a person who has the special task of founding and establishing churches apostle, messenger (of God) (EP 2.20); (c) especially of the Twelve chosen by Jesus apostle (LU 6.13)

This is the role taken up by many today and in some churches they are called apostles but in most western churches they're called missionaries. They're specifically sent with authority and their purpose goes beyond preaching the gospel. They are also expected to start churches and eventually move on leaving local people in charge as pastors, elders, teachers, etc. They often stay in communication with these churches and correct doctrine or deal with bigger internal issues that may arise the local church cannot or doesn't know how to handle.

Evangelist - Friberg Lexicon 11967 one who brings or announces good news; in the NT evangelist, preacher, or teacher of the gospel (AC 21.8)

This role is seen in a few forms. There are traveling preachers who go from church to church or town to town and preach. Those usually don't stay for a very long time. It could also be someone that preaches around their town on street corners who would point people interested to an already established local church. They can also be someone who fulfills only the preaching role in a local church and are not involved directly in many/any pastoral functions, many of the very large churches have this situation. Billy Graham is an example of an Evangelist.

The second question: In regard to churches and traditions that affirm the office of apostle is still distributed in the church today, how do they view the difference between the office of apostle and evangelist?

The only Christian Churches that I've known of to have an office of apostle(s) are within the Pentecostal denomination. The role varies quite a bit from church to church or from one sub-denomination to another.

Apostle - One called directly by Christ

This is the one item that is the most in common. That all the apostles are called directly for a mission or purpose that is lifelong. In most cases the office is much more similar to that of an elder or bishop that exercises authority and corrects doctrine. They may or may not have ever established a new church. They may have authority over a single church, multiple churches or an entire sub-denomination.

Evangelist - Anyone who proclaims the gospel

The evangelists range from itinerant preacher to everyday life gospel proclaimer to motivational speaker. The office itself does not have any authoritative position although many may have more than one gift and also have a role like that of a pastor.

From a Pentecostal perspective that believes in a restored five-fold office-gift ministry:

An apostle is a "Sent One" & and an apostle's "sentiness" manifests in the supernatural authority they bear in terms of enacting the mission that they have been entrusted with. Apostles are leaders par excellence, and are usually identified after they have pioneered the establishment of new networks of churches. They tend to be very focused on discipling and raising up effective leaders who can further multiply disciples and oversee congregations or ministries. They excel in imparting the vision they carry to others. While church planters and missionaries may be engaged in "apostolic work", they tend not to be recognized as apostles until they have successfully overseen the establishment of many individual congregations.

It could be argued that evangelists are also "sent ones" as they have received a burden from the Lord to go out and win souls at every opportunity. While apostles may also be effective soul winners, they are distinguished from evangelists by their grasp of the bigger picture - how the whole church can function effectively and grow in maturity; an evangelist on the other hand primarily concerns themself in conversion of the lost, content to leave ongoing discipleship and church governmental issues in the hands of others, although the most effective evangelists are usually actively involved in discipling, raising up and empowering more evangelists.

Examples of people regarded as modern day apostles are Chuck Smith and Phil Pringle.

Examples of people regarded as modern day (office-gift) evangelists are Billy Graham and Reinhard Bonnke

This perspective retrospectively applied to the interesting case of the two principal leaders in the (first) Methodist revival - George Whitfield, the greatest soul-winner of his generation is rightly acknowledged as an office-gift evangelist; whereas John Wesley although a very gifted soul-winner as well, possessed a superior organisational ability that marks him as the apostle of Methodism. While the two had significant theological differences (Calvinism vs Arminianism), the former acknowledged the wisdom of the latter's establishment of his "classes" (effectively small group discipleship meetings) in these terms:

Whitefield met an old friend, Mr John Pool and accosted him in the following manner:

“Well, John, art thou still a Wesleyan?”

Pool replied, “Yes, sir, and I thank God that I have the privilege of being in connection with him, and one of his preachers.”

“John,” said Whitefield, “thou art in the right place. My brother Wesley acted wisely—the souls that were awakened under his ministry he joined in class, and thus preserved the fruits of his labor. This I neglected, and my people are a rope of sand.” (source)