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.