Why is it that in Europe, most written languages (Spanish, English, German, ect.) use very similar alphabets (Latin), while countries in South East Asia, which are similarly geographically close to one another as in Europe, use vastly different alphabets from one another (Thai, Mandarin, Japanese, Sanskrit, ect.)?
The answer to this question is fairly straightforward along the lines of historical developments and cultural influences and goes pretty much along the lines offered by jamesqf's answer.
However, a much more illuminating answer would focus on the problematic nature of the assumptions behind the question.
First, Europe and South East Asia are not necessarily the best categories for this sort of comparison.
When it comes to writing systems, you may be better off talking about Mediterranean-Atlantic cultural sphere. Here, you will find at least five different (albeit related) writing systems: Latin, Greek, Cyrillic, Arabic, Hebrew currently in wide usage (millions of users across multiple languages). They can all trace their origins to similar roots but ultimately they underwent independent processes of development and cultural spread.
On the other side you have at least two historical spheres of cultural influence: Brahmic (Indian languages, Tibetan, Thai, Lao) and Sinic (Chinese, Japanese, Korean, Vietnamese). So, talking just about South East Asia as one area is misleading.
If you only focus on the artificial concept of 'modern Europe', you will find that the prevalence of the Latin script is due mostly to its mono-cultural origins of learning in Western Christendom. What may now seem like some sort of inevitable default state is, in fact, a consequence of Europe being a cultural and technological backwater for well over a thousand years while other places around the world flourished and promoted the development of more or less independent cultural traditions strong enough to carry their own literacy. The surface dominance of the Latin alphabet is only about 200 years old coming on the heels of mass literacy in the West spurred by industrial revolution (to simplify things greatly).
Most if not all the area of Europe which uses the Latin alphabet was once part of the Roman Empire, and used Latin as the language of trade even if it wasn't the native language (much like English today). Many European languages (the Romance family) are direct descendants of Latin, so not surprising they'd keep the alphabet as well.
After the Empire fell, the Roman Christians used Latin, so it eventually became the language of scholarship, pushing out local writing systems (e.g. Norse, Irish).
China and India are in fact quite isolated from each other. While I'm no expert (and welcome correction), what I've read suggests that there was more contact between Greece/Rome and India, than between India and China.
Japanese does use Chinese characters (kanji) in its writing. Because the language is very different - lots of inflected endings for tenses &c - they added kana, so each Japanese word is typically kanji+kana. (I think Korean is similar.)
I don't know that much about south Asian linguistics, mostly European and Eastern Asian, so my answer will be limited but I hope helpful given those limitations.
The most important thing to remember when comparing eastern vs western script is that for a long, long ways back, European script was phonetic, whereas Chinese script and such was not. By this, I mean that alphabets such as Greek, Egyptian, Hebrew, and Latin were meant to represent sounds, which were combined to represent words. What this meant, and why this is significant, is that it means they are easily interchangeable. The Rosetta stone is an example of how easy it is to crack multiple phonetic alphabets. Each phonetic language has a symbol for the "k" sound (though some have "kh" differentiated). In addition to being interchangeable, this made it far easier to adapt a certain alphabet to different languages. The Catholic Church in particular had a resounding impact in this; they made it their mission to provide the Bible in numerous languages. Sometimes, they displaced preexisting scripts entirely; Germanic runes were replaced for the much more fluid Latin script. Other times, alphabets were made entirely from scratch by the church; the single greatest undertaking of this was the creation of Cyrillic script from the Slavic languages, when the Orthodox church requested that the church help them formulate a written means of communication, and two priests by the name of Methodius and Cyril formed an entirely new script composed of Hebrew, Greek, and Latin letters and suited for the consonent and trill heavy Slavic tongue. So to answer the European side, it started out diverse, but the Church used its influence to push a very efficient script capable of suiting multiple languages throughout the various lands and nationalities.
Eastern Asia, on the other hand, has traditionally followed a very different form; their script, instead of focusing on somatic components, instead were meant to represent individual ideas and concepts and words. So instead of having six letters for "finger," they would have one letter for "finger." The actual script themselves were often meant to relate to the idea, much like the formations of stars are meant to depict constellations. This gets cluttered, however, when you consider two things: 1) it is not easily interchangeable, and 2) multiple lands were doing this simultaneously. Regional divergence was highly significant, and is the reason why there are so many characters that can represent the same idea. It was also complex and difficult to teach; although certain symbols were universally used such as markings for roads or types of buildings, most of the more abstract concepts of actually writing were essentially impossible to incorporate into the general population. It got to the point that the Korean king Sejong the Great had some of the best minds in his kingdom create a new, somatic alphabet, one that is commonly esteemed to be the most phonetically "accurate" because the characters are not only phonetically representative, but are meant to represent the positions of the tongue and throat while making the sound. Another nation, Japan, also eventually adopted another alphabet following somatic components, as well as adding on a third alphabet later on typically used to represent Western concepts, giving Japan a total of three modern working languages. China followed similar routes as well. So the basis for alphabetic confusion in Eastern Asia is basically that it started as a conceptually similar use of a non-phonetic alphabet, which is very vulnerable to regional divergence, and different lands used different means to eventually cope with this and change it to a regionally acceptable phonetic alphabet.
To add a little to Sanskrit and Indochinese scripts, though, although I can't go into detail on their development, it is important to point out that despite your assumption, they were not geographically close. India and the main parts of China were significantly seperated, by distance, jungle, and massive mountain peaks. Indochina was isolated by extensive jungle regions not suitable for agrarian societies. The expectation that they would have linguistically similar origins are slim at best.
It more or less a historical artefact the Europe has so few writing systems. At the turn from the medieval age to modern era, most European languages had developed their own style of writing the Latin alphabet and the difference in style can be seen in the typefaces for early prints. Some of these differences survived until today where an uncial style is associated with Ireland and fraktur with Germany. But somehow those differences levelled out and weren't seen as anything but differences in style.
On the other side, all the scripts of India derived from the Brahmi script were considered separate scripts despite sharing the same order of the syllabary, the same set of basic letters, and the same principles of forming syllables.
The Chinese, Japanese, and Korean script are independent developments from that and are unrelated to the Indic scripts.
I'll try to answer this from Indian perspective. In ancient India (the Vedic Period), Sanskrit has been the lingua-franca. All our religious texts including the Vedas, Upanishads, Ramayana, Mahabharata, etc. have been written in Sanskrit language (the Devanagari script). And India in those times, extended upto Persia in the West and the Himalayas in the east. Some of you were right in pointing out that we had more interaction with Greece and Persia during ancient times than China. The reason being, the Himalaya mountains which acted as a great geographical barrier between India and China.
Since there were no such major obstacles between Greece and India, trade and information exchange was possible. Alexander invaded India just before Chandragupta Maurya established the Maurya empire around 320 B.C. During the Maurya period, the Pali language was gaining more and more importance which was nothing more than a corrupted version of Sanskrit. Indeed, most of the Buddhist texts were originally written in Pali and not Sanksrit.
However, when you move towards South India, you will find totally different languages (Tamil, Telugu, Malayalam and Kannada) which are written in their own scripts and have no link whatsoever to Sanskrit. The reason for that is speculative, though some scholars suggest the Indo Aryan migration hypothesis.