There are obvious differences between the world we live in, and that in which the Bible was written. These differences are cultural, technological, and practical, plus it seems evident that the early church had more grace and power than we typically experience. Reading the book of Acts makes us feel far from being like the early Church, and yet the Bible is clear that 'The glory of this present house will be greater than the glory of the former house,’ says the Lord Almighty. (Haggai 2:9) When Jesus conducted His first miracle in Cana the wedding host said "Everyone brings out the choice wine first and then the cheaper wine after the guests have had too much to drink; but you have saved the best till now." (John 2:10)
God likes to do His best things last. He doesn't start with His best thing, and then do His second best, until finally all He has are options of diminishing quality. The Body of Christ in the world today is supposed to supercede the Church in the book of Acts. This we have not yet done, and apostles are needed to help us get there.
Right now God is restoring apostles, but they don't yet have their full authority. A fuller understanding of who apostles are, and what they do has not arrived, and even those that have been called, may not understand it themselves. There is much to learn, and a journey still to take.
With the restoring of apostles, there is still much difference between what we see, (or think we see) in scripture and what we see (or appear to) see today. Part of this difference is due to the unfolding understanding, and part due to the differences already mentioned above.
So What Are The Differences?
Modern day apostles do not yet posses the freedom and the authority that Biblical Apostles had, but they will in time.
Modern day apostles don't effectively govern churches today, but this is where Christ is taking it.
Modern day apostles are yet to really find their place in the Body of Christ.
Many modern day "apostles" have not been tested to be apostles, and some of them in fact aren't apostles at all. There doesn't seem to be an adequate practical and widely understood way of how to test or approve of apostles yet.
Modern day apostles may not "seem" like their biblical counterparts because the everyday life aspect of Biblical apostles is not mentioned much in scripture. The biblical apostles often seem more 'super' than modern day apostles.
Modern Day apostles have conveniences available to them that their Biblical counterparts didn't have such as technology, world travel and common languages.
Not all modern day apostles are going to be like St Paul. It is not church planting and travelling which determine apostolic gifting. Anyone can travel and plant churches.
There are some similarities between Biblical and Modern day apostles too. They should both be walking with Christ, know the mind of the Lord, be willing to serve and go wherever directed. The apostle is a "sent one" (apostolos) which in Greek means “to be sent” - an ambassador for Christ. Christ should still command His apostles and they should still listen.
Though times, languages, clothing and technology will change, one thing should remain. This is the heart of God in the apostle. The apostle should be gentle, humble, and obedient to Christ. The apostle should love, and in doing so teach the believers how to love too.