The Byzantines regarded themselves as being above the "barbarians" from the west (nicely portrayed in the Alexiad by Anna Comnena). Relations were also pretty strained after Charlemagne was crowned Emperor in Rome, since the Eastern Roman Emperors claimed to be and styled themselves the only rightful successors to the Roman Empire. Since Roman and eastern churches were drifting apart more and more, and cultural differences were pretty great, Byzantines were often seen as outsiders and the Eastern Roman Empire scornfully referred to as the Empire of the Greeks. It all culminated in the sack of Constantinople during the Fourth Crusade and the creation of the Latin Empire by Western Europeans.

I think we should be careful not to overgeneralize about the "West" as a homogenous unit. Even on the Italian Peninsula, we can see variation in relations between the Byzantines and the “West.” For centuries, Venice and the Byzantine remained on friendly terms and functioned as trading partners in the Mediterranean. Venice was actually initially a Byzantine colony/outpost, but over time its power increased and the nature of their relationship changed. The proliferation of Venetian merchants in Byzantine cities during the central Middle Ages created tensions between the two countries and it was a Venetian navy that sacked the capital of Byzantium in 1453. Donald Nicol’s book Byzantium and Venice: A Study in Diplomatic and Cultural Relations considers this topic in depth.

Another fascinating case study is the Byzantines and the Normans. In Anna Comnena’s Alexiad, we can see scathing depictions of the Normans and their crusading leaders - Bohemond of Antioch and Tancred. Indeed, the Normans of southern Italy/Sicily had waged wars against the Byzantines in the years leading up to the First Crusade, which created a complex diplomatic situation when Bohemond and Tancred turned up with a crusader army created (ostensibly) to retake lands for the Byzantine Empire. Granted, Bohemond and Tancred clearly had different motivations in mind for their crusade as seen in their actions in Cilician Armenia and Antioch, but it does give an indication of the tenuous relationship between the two groups.

In the context of the First Crusade, we can also consider the Papacy. In most of the surviving copies of his speech at Clermont in 1095, Pope Urban II spoke of the need for the nobility of Western Europe to defend their Christian brothers, i.e. the Byzantines. Bear in mind that this talk of Christian brotherhood is coming only some forty years after the Great Schism (1054) between the Latin and Orthodox Churches. Given the immense popular response to the First Crusade, it is reasonable to argue that there was at least some pro-Byzantine sentiment in Western Europe. To be fair, I should clarify that there were a plethora of motivations for crusaders to take up the cross - the desire to defend the Byzantine Empire was one of many.

These are just two examples of the varied relationships between Byzantium and the powers on the Italian Peninsula. One could write for hundreds of pages on other groups like the Genoese, Lombards, and Pisans in the context of their relations with the Byzantines. If you are interested in the topic, I would recommend choosing a particular time period and group to study in relation to the Byzantines. Depending on the time and place, perceptions on both sides of the coin will vary.