How are the characteristics of the Roman Empire different from the Roman Republic?

This is a
great question. The first point we need to make is that Rome changed under the Republic and the
Empire quite a bit. Therefore, we need to be flexible and speak in terms of the big picture. If
we do this, here is what we have. 

First, the Empire had an emperor. This is,
undoubtedly, the most distinguishing mark of the Empire and what sets it apart from the
Republic. The Republic had consuls, praetors, and quaestors, to name a few of the magistrates.
Moreover, there was always a number of them, so that power was diffused. In the Empire, we still
had these magistrates, but there was no question who had power - the Emperor. 

Second, In the later Empire, post-Diocletian, the Roman Empire was divided into the
West and the East. This might seem like a small thing, but this is an important point, because
the Empire would develop on two tracks. In fact, under Constantine, a new city was built, which
will survive until 1453 - Constantinople. 

Finally, I would say that the
Roman Empire by the time of Constantine changed religiously. The Empire embraced Christianity
and this would spread to Constantinople as well. From Rome and Constantinople, unwittingly, Rome
gave birth to Roman Catholicism and East Orthodoxy.