What's the difference between Christianity and Orthodox Christianity?

Orthodox Christianity maintains the whole teaching of the Apostles and also maintains Apostolic succession, which means each Bishop of the church today can trace their position back to one (or more, usually more than one) of the 12 Apostles. 

Before the present time, this was not really obvious, but today we have so much writing from the church fathers available in a variety of languages and so it is totally provable.  We can make a list of beliefs in which the early church differed from forms of Christianity today like Protestants and Catholics, and in every one of those cases the Orthodox Church matches that of the early church.

These include items like the real presence of Christ in the Eucharist, the Sacraments of the Church called Holy Mysteries, and elements of everyday Christian practice like Fasting and having a daily prayer rule.

In terms of worship, the main service of Orthodox Christians is called Holy Liturgy.  This is, in the opinion of many people, much more worshipful than an ordinary Protestant service, and contains much more collective prayer. 

The Eucharist, one of the Holy Mysteries (called 'Sacraments' in the Roman church) is given each week, and Orthodoxy teaches that Christ Himself is really present in the bread and the wine, which are called the Body and Blood.  This is why you see people bowing down or prostrating towards the cup, and why in some churches you see people reach out and touch the robes of the priest who is carrying it. (edited)

We teach these things because it is what Christ taught, and what the Apostles taught. It is thoroughly supported by scripture that Christ is really present in the Eucharist, but we are very comfortable leaving the mode of his presence unexplained - a "Holy Mystery" of the church. 

As the Psalm says, "God is glorified in His saints".  When we praise them, we praise the good they have done, and we understand very clearly that all good things come from God.

We know that they are not dead, but alive with God in that place that our Lord said He has gone to prepare rooms (John 14:1-4).  It glorifies God and pleases Him to do wonders through His saints.

This is why we pray to the saints and to Mary, who is called "the Holy Theotokos", which translates to something like "birth-giver of God".  We do not worship them.  We venerate them and honour then, even bowing to them.  But, and this is critical, we understand very clearly that whatever glory they have is "borrowed".  

Regarding bowing towards people, there's a quote from one of the saints that says: "If I were walking after Liturgy and I saw a man and an angel coming towards me, I would sooner bow towards the man than the angel, because the man carries Christ inside him, and the Angel does not."