The problem with Greece is that its history is regarded as fundamental to European and, by extension, Western civilization. Particular elements of Western culture, such as philosophy and democracy, are thought to have been invented in Greece and stand at the root of European/Western culture. More or less at random, I’ve picked the Cambridge Illustrated History of Ancient Greece (1998), edited by Paul Cartledge, from my shelf, and this is a passage from the back cover blurb:

Suppose you could book passage on a time-machine and have yourself transported back to Greece during the fifth century BCE. In Athens alone you might encounter Aeschylus, Alcibiades, Arisophanes, Herodotus, Plato, Socrates, Sophocles, Thucydides, and Xenophon (to name but a few) – all of them contributing to the enmorous energy released in this small hothouse of culture and politics which generated the political, artistic, cultural, educational, philosophical and scientific groundwork on which so much of Western civilization and culture is based. From the first democrats to the first person to propose the theory of atoms (2500 years ago), the impact of ancient Greece has been profound.

As a result, Greece is thought to be distinct from the ancient Near East, and this notion has been extended to include the pre-Classical Aegean. This came about quite naturally since the earliest explorers of the Aegean Bronze Age – most notably Heinrich Schliemann – believed that the world of the Mycenaeans corresponded to the epic world described by Homer in his Iliad and Odyssey.

In reality, of course, the idea that the Greeks (and their heirs, the Romans) were at the cornerstone of European/Western civilization dates back to the Renaissance, when artists, writers, and other intellectuals actively sought to latch onto the Classical heritage as a way to distance themselves from what scholars at the time termed the ‘middle ages’, the period between antiquity and modernity. For a fun introduction to the topic, I recommend Michael Shanks’s Classical Archaeology of Greece: Experience of the Discipline (1996).

Naturally, we cannot leave the ancient Greeks entirely blameless in this. The Greeks were the first to give the European continent its name, since Zeus – in the form of a white bull – abducted the Phoenician princess Europe from the continent of Asia. Herodotus, in the opening book of his Histories, spends some time to recount the various instances of tit-for-that raiding of European women by Asians and vice versa, including the Trojan prince (Asian) Paris’ abduction of the Spartan (Greek/European) Helen, and so on and so forth. According to Herodotus, the Graeco-Persian Wars of the early fifth century BC were only the latest conflict between Europe and Asia.

While Herodotus wrote the Histories to record the heroic deeds of both Greeks and Asians, as specified in his introduction, some modern commentators have nevertheless tried to exaggerate the differences between East and West. Hence, a book like Victor Davis Hanson’s The Western Way of War: Infantry Battle in Classical Greece (1989) contrasts Greek/European modes of warfare (open and honest) with Persian/Asian (cowardly, reliance hit-and-run tactics). In truth, of course, such dichotomies are not particularly useful, and they tend to play down similarities. In my opinion, works like that of Hanson do more harm than good, but that’s an entirely other issue that I won’t go into here.

In practice, it’s difficult to pinpoint where ‘Aegean/Greek culture’ ends and the ‘Near East’ begins. There is a lot of overlap. For the historical period, for example, it’s clear that many Greek cities in Asia Minor and on the islands off the coast had closer ties with Anatolia than with mainland Greece, such as Lesbos: see e.g. A. Dale, ‘Alcaeus on the career of Myrsilos: Greeks, Lydians and Luwians at the East Aegean-West Anatolian interface’, Journal of Hellenic Studies 131 (2011), esp. pp. 21–23 and 23 n. 43. In ancient Greek art history, material culture of the seventh century BC is so heavily influenced by Near-Eastern art that we speak of an ‘Orientalizing’ period.

Things become even more complicated when we try to take the Greek world in a broader sense into account, which ranged from Greek cities and outposts in France and Spain to the Levant and the Black Sea region. It’s often very difficult, if not impossible, to tie ethnic identity to material culture. See, for example, Jonathan Hall’s Ethnic Identity in Greek Antiquity (1997) as a good introduction on that.

In the Bronze Age, there are many connections between the Aegean and the Near East. Like in the Near East, Minoan and Mycenaean culture appears to have centred, in the Middle and Late Bronze Age, on the palace, which formed the centre of a redistribution economy (that we know now, for both the Aegean and the Near East, to have been more specialized than was often assumed in the past). Both cultures appear to have been ruled by kings (assumed for Minoans, proven for Mycenaeans on the basis of Linear B tablets), similar to the situation in the ancient Near East. Art shows Near-Eastern influences, such as the inclusion of fabulous creatures like griffins and sphinxes. For a good introduction, see Oliver Dickinson’s The Aegean Bronze Age (1994). For something more recent and more detailed (and thus less easy to read), refer to The Cambridge Companion to the Aegean Bronze Age (2008), edited by Cynthia Shelmerdine.

Wall-paintings, among other things, attest that Minoans and Mycenaeans visited Egypt. We also know that the Mycenaeans were active in Anatolia and that the Hittite King, at least, addressed one Mycenaean king – a ruler of a region known from the letters as ‘Ahhiyawa’, possibly to be rendered as Homeric ‘Achaea’ – as ‘Great King’. The latter was an appellation reserved solely as a form of address among the rulers of the major Near-Eastern kingdoms (Egypt, Babylon, Assyria/Mitanni, and the Hittites). This suggests that the king of Ahhiyawa, a Mycenaean, was considered a powerful figure by the Hittites. You can read more about these letters in Harry A. Hoffner Jr’s Letters from the Hittite Kingdom (2009).

So the interconnections between the Aegean and Near East are complex. They engaged in trade, diplomacy, and warfare, and they influenced each other as far as material culture was concerned. Contacts were facilitated because the elites, at least, shared a common way of doing things: exchanging gifts, entertaining guests, drinking alcohol, and so forth. While there were many differences between the Aegean and the Near East, there were also lots of similarities, and it’s difficult, perhaps impossible, to draw any hard dividing lines between one and the other.

Hopefully this answers your question somewhat. Feel free to post follow-up questions.

Mycenaean Greece was distinct from Near Eastern powers, but contact was regular and frequent. Tin for bronze came from Afghanistan, and prestige items from the Near East have been found in Mycenaean palaces (notably the lapis lazuli cylinder seals found at Thebes).

More importantly are the Hittite Ahhiyawa texts, which most scholars accept as referring to the Mycenaeans (a Hittite form of the word 'Achaea'). These texts are an indicator that there was formal recognition of a Mycenaean king and power outside of Hittite control (although all of Mycenaean Greece should not be considered a unified power).

The Ulu Burun shipwreck provides additional evidence of regular contact. They were carrying weights and measures and trade goods that would've been relevant to the entire region.

So there's a good deal of trade contact, a shared poetic tradition (Mycenaean Greek and Hittite are both part of the Indo-European language family, and there are elements of a consistent IE poetic tradition), and diplomatic ties.

But one would be hard-pressed to argue against the autonomy of Mycenaean kingdoms vis a vis the Near East. The Mycenaean texts say nothing at all about trade or external contact.

As a supplement to /u/shubahastorn, there was an "International Style" in the Late Bronze Age which is detectable in the products of many LBA cultures across in the Near East and the Mediterranean. Marian Feldman wrote a book on it:

Art and international relations during the Late Bronze Age formed a symbiosis as expanded travel and written communications fostered unprecedented cultural exchange across the Mediterranean. Diplomacy in these new political and imperial relationships was often maintained through the exchange of lavish art objects and luxury goods. The items bestowed during this time shared a repertoire of imagery that modern scholars call the first International Style in the history of art. Marian Feldman's Diplomacy by Design examines the profound connection between art produced during this period and its social context, revealing inanimate objects as catalysts - or even participants - in human dynamics. Feldman's fascinating study shows the ways in which the exchange of these works of art actively mediated and strengthened political relations, intercultural interactions, and economic negotiations. Previous studies of this international style have focused almost exclusively on stylistic attribution at the expense of social contextualization. Written by a specialist in ancient Near Eastern art and archaeology who has excavated and traveled extensively in this area of the world, Diplomacy by Design provides a much broader consideration of the symbolic power of material culture and its centrality in the construction of human relations.