Why was Europe so technologically and culturally advanced?

First of all this was not always the case in history. In the period approximately 400-1500, European populations were not more technologically advanced than some populations in Asia. And even before that, most of Europe, except a narrow strip near the Mediterranean was in quite primitive state with respect to science and technology. And the main place where science developed in Antiquity was Alexandria (in modern Egypt).

Science and technology begin to develop intensively in the Western Europe in XVI century, and it is difficult to name one reason for this, but perhaps one of the main reasons is capitalism, and in general, the social environment which made this development possible.

For example, paper, movable print and possibly firearms were invented in China, earlier than in Europe, but invention of paper and print did not trigger a revolution in China of the kind that happened in Europe. Mass production of printed books did not start there. And firearms as soon as they penetrated, or were invented in Europe, very quickly reached higher degree of perfection there. Another example is the invention of steam turbine in the Hellenistic Alexandria. They invented it, but what use did they make of it? These are just three examples, that proper social conditions are crucial for development of science and technology.

Of course, one can ask why these social and economic situation developed in Europe and not elsewhere, but to this there is probably no short answer. One has to analyse the whole history of Europe and other parts of the world to see why it happened in Europe.

It is not true that "Europeans used money while other people used barter". Money was quite common in many places.

On the last question. It is true that Greece was the birthplace of science. But not the only birthplace. Astronomy, for example started in Babylonia. This is not true about technology. All sorts of technology where invented in many places, certainly Greece in not "the birthplace of technology".

In the Hellenistic states (NOT in Greece itself, but on the territories of modern Egypt, Turkey and other parts of Mediterranean) science reached very high degree of development and started to influence technology. But this was a relatively short period, about 200 years, approximately from the time of Macedonian conquest to the conquest of these states by the Roman empire. The Romans were not interested in science, and mostly employed the Greek engineers for technology. Even the severe shortage of the engineers in the empire did not cause any form of engineering education. The crucial thing is that scientists and engineers has very low social status in the Roman empire. Few centuries after that, the development of science in Europe stopped completely, and there was a strong regress in technology.

The Hellenistic science partially survived in the East (Persia, Muslim countries, Byzantia, even in India). Then in XVI century, all this had to be reborn in Europe.

EDIT. I anticipated that my claim that "science was born in ancient Greece" will raise objections. So let me explain. It is a common misunderstanding that "Greeks did not do real science because they did not make experiments". First of all, not every science requires experiments (astronomy, geography requires only observations. And certainly Greeks and Babylonians did and recorded observations). Second, Hellenistic scientists DID do experiments. And not only Archimedes. They created several parts of physics which was closely connected with engineering, namely statics, including hydrostatics, pneumatics, geometric optics, geodesy. Ptolemy described his experiments measuring refraction, for example. Hero of Alexandria constructed automatons and steam engines. Ctezibius constructed water clocks and water organ, etc. A lot of science went to construction of artillery, and not only Archimedes was doing that.

EDIT2. It is common to call Euclid, Hipparchus, Ptolemy and Diophantus "Greeks". They indeed wrote in Greek, and their names sound Greek, but they did not work in Europe. Concerning their "ethnic origin" we have no information at all.