While the tradition in Greek Christian mysticism was one concerned with metaphysics and structure, the Latin tradition tended to be more personal. Augustine belongs to this tradition, but he deserves his own heading.
History: In about 303, Diocletian was the emperor. He was the last great persecutor of Christians. By 333, Constantine had had his vision of the cross on the battle-field. By 378 under Theodosius, the empire was solidly Christian.
At the same time, though, the empire is split into east and west. The empire was more often ruled under two leaders in the 4th century than under one. There was a commensurate split in the church. Rome was still the head of the west, and Constantinople became the head of the east. In terms of the church, though, there were several equally important centres in the east: Constantinople, Jerusalem, Antioch, Alexandria. Rome asserts its primacy over time (for instance, at the First Council of Constantinople, 381). While the bishops of Rome did not hold great authority at the beginning (being overshadowed by people like Ambrose and Augustine), with Leo I (440-454) there is an emphasis on the authority of Rome.
The churches in the east and west took different directions. There were monastic movements in both, for instance, but they differed. Monasticism in the west was a more communal sort of effort (possibly owing to the harsher climate). The solitary ascetic monk remained the ideal, but rarely the reality. The monastic orders came not from the common people, but from the educated. The Latin mystical tradition can be seen as rooted in this early monastic life.
Much of this monasticism came intermingled with neo-Platonism. Plotinus was translated into Latin by (among others) Marius Victorinus (Augustine describes him in the Confessions, Book 8). It is important to recognize that, with the split in the Roman Empire, Greek became all but unknown in the west. Augustine knew a bit, but not much. So, Plotinus was mediated to the Christian West by a few scholars.
One of the ideals of the monastic community, and one which informed early Latin mysticism, was the ideal of virginity. Christians had been fascinated with this for a long time, but in the 4th century leaders became convinced that virginity was an inherently higher form of life both in theory and in practice. It is the beginning of the belief in the perpetual virginity of Mary, for instance. This opinion was not universally held, it should be noted, but the opposition could not stand against the opinions of those like Ambrose and Augustine. Despite this, most clergy continued to marry; it was only in the 11 century that celebacy for priests became mandatory.