An apostle then and now
How does the title apostle apply in biblical times and how does it apply today?
The Twelve (as sent to the Jews) and Paul (as the apostle to the Gentiles, Galatians 2) are strictly speaking the apostles of Jesus Christ, specifically called and commissioned by him (Matthew 10;1-4; cf. Acts 1:15-26).
They serve as the foundation of the church since they were entrusted with giving the church of all generations the sound doctrine by which the church lives and thrives (Ephesians 2).
They were qualified for the apostolate by being called directly by Jesus Christ ("An apostle, not of men, neither by men, but by Jesus Christ," Galatians 1:1; John 6:70; 15:16); they saw the Lord physically as risen from the dead (1 Corinthians 9:1; Acts 10:40-41); they were conscious and knew that they were endowed with a special authority (1 Corinthians 2:13; 14:37-38; they had the insignia of the apostles, that is, they were enabled to perform signs and wonders, in confirmation of the gospel which they proclaimed (2 Corinthians 12:12; Romans 15:18-19; Acts 5:12); they were also profusely blessed in their labors (1 Corinthians 9:1). Thus it appears that it is quite impossible for them to have successors.
The spurious claim of the Catholic Church is laid bare since Roman bishops (and nobody else for that matter) are seen to be unqualified for the apostolate. The simple fact of the matter is that a foundation is laid only once, at the beginning, and that is what the original apostles were (Ephesians 2).
To give the whole picture, though, we observe also that some men in apostolic times were designated "apostles." But with a mighty difference. For basically the word apostolos simply means a sent one, derived from the verb apostello, meaning to send. Now anyone sent from a local church and by a local church for the fulfillment of some mission or service, was called an apostle, but in no sense meaning an apostle of the same caliber or authority as the Twelve and Paul.
Thus, Paul's companion in his missionary travels, Barnabas, was called an apostle, for the simple reason that both of them were sent out from the Antioch church. In this secondary sense also Epaphroditus is described as an apostle.
There is no harm, and indeed I would encourage the use of the word apostle for today as long as we keep this distinction in mind. The Neo-Apostolic church errs grievously in claiming (similarly to the Roman Catholic Church) that it enjoys the ministry of apostles of the same kind and nature as were the twelve. This implies continuing revelation, which is the hallmark of every sect and cult.