I've read everything i could about history since i was 10 or 11, but coming from Germany, i never encountered the Byzantine Empire until i was an adult; i remember seeing them in Age of Empires and didn't know what they were supposed to be. In the German historiography, you either have Charlemagne as the "real" emperor (plus the Holy Roman Emperors, but that's a different story), or the Roman Empire just ceded in 476 altogether. I'm active in some english-speaking forums (fori?) for almost 10 years now, and that's where i encountered the anglo-saxon historiography. The way i see it, the Byzantine Empire is either the Roman Empire itself or it's real successor there, while the HRE is just an impostor at best. Now that i don't have the adequate literature for that, i never found any hints for my question; the byzantines called themselves emperors (or Augustus, for that matter), and so did the ottoman sultans, starting with Mehmed the conquerer (Kayser-i Rum). I don't see a reason to say that the Roman Empire ended in 1453, you might as well say that the dynasty, the language and religion changed, like it already did once: when the capital was shifted to Constantiople. My best guess is a racial or anti-islam bias, but that's hardly fair. I hope i don't sound rightous about german historiography, because i sure as hell don't want to start a flamewar.
There is a high degree of continuity between the Late Roman Empire and the early 'Byzantine' Empire. The Empire was already effectively split into two halves, the Eastern half was generally considered more important and more prosperous, and the centre of power was already in the East. When the West 'fell' no one would have thought, "oh, that's the end of the Roman Empire". It was a significant loss of territory and a sign of the diminishing prospects, but they still considered themselves Romans, and the Roman Empire. There was no upheaval, no regime change, and only a slow process of transition and cultural change that created a distinct entity that we usefully call 'Byzantine' mainly for chronological purposes.
Not so with the Ottomans. The Ottoman state existed long before it conquered Constantinople, it had an independent and unrelated history as a socio-cultural and political entity. Conquering the last of the Byzantine territories meant, as you say, massive changes in ruling dynasty, ruling language, and dominant religion, at which point you can indeed say that the independent existence of the Roman Empire came to an end. The Ottomans did not maintain or co-opt the Imperial state apparatus, the conquered.
This is, essentially, why the Eastern Empire is considered a genuine succession of the Roman Empire, and the Ottomans are not.
There is a big difference between the Byzantine Empire and the empires of the Ottomans and of Charlemagne: it did never start of as seperate entitiy that at one point in time started to claim the Roman imperial title. The Frankish kingdom of Charlemagne had already a long history when its king was crowned emperor by the Pope in 800 A.D. The Ottoman familiy was ruling their sultanate for several generations when Mehmed II. added the title of Kayser-i Rum to the other titles he already claimed. And afterwards both still considered themselves first and foremost as the frankish king/muslim sultan. They now just claimed the additional title of emperor.
With the Byzantines you have no such clear point in time when they start to claim the roman legacy. There was no foreign invasion by "Byzantines" that conquered the Roman Empire. Those Byzantines simply are the Romans of Late Antiquity and the Middle Ages living in the same empire as before that just has shrunken down considerably from the 5th to the 7th century. That's why historians can't even agree when Byzantine history actually starts. The empire was never officially founded as a seperate state. You could argue for the foundation of Constantinople in 324/330, the final partition of the Roman Empire in 395, the end of the Western Roman Empire in 476/480, the reign of Justinian I. (527-565) or the reforms Herakleios at the beginning of the 7th century as a starting point and you never would be wrong per se. But you have to keep in mind that the question about the start of the Byzantine Empire is only asking about the point in time when the Romans finally became so different from what we normaly would consider "Roman" that we have to give them another name.
As for your point about German historiography, I can confirm that Byzantium plays almost no role at all in german school education. But I'm not really shure if Germany is the exeption in this or only following a worldwide trend. And there certainly are some good books in German about Byzantium. Try for example the works of Ralph-Johannes Lilie, one of the foremost Byzantine historians in Germany. Byzanz: Geschichte des oströmischen Reiches 326–1453 (München 2010) is a fine little introduction from the Beck'sche Reihe. If you want something more extensive you can read Byzanz. Das zweite Rom (Berlin 2003) from the same author. Then there is Georg Ostrogorsky's Byzantinische Geschichte 324 bis 1453 (München 2006) which isn't really totally up to date but still a very good overview. If you want something that doesn't only focus on the history you can read Claudia Sode - Mabi Angar, Byzanz. Ein Schnellkurs (Köln 2010) or Andreas Külzer, Byzanz (Stuttgart 2012).
I answered this question a little while back. You can read it here.
Feel free to follow up with any questions you might have.
The Roman Empire had divided into two by 395: the Western Roman Empire and the Eastern Roman Empire. The Roman Empire did fall by 476, as you said, but this was only when Romulus Augustulus, the last Western Roman Emperor, was deposed. Meanwhile, the Eastern Roman Empire still existed, and would exist until 1453 (except for the interlude of the 4th Crusade). In Western historiography, the name "Byzantine Empire" is used to describe the Eastern Roman Empire. The Byzantine Empire therefore is not really even a successor to the Roman Empire but the Roman Empire itself, albeit one half of it.
It could be argued that the Holy Roman Empire was a political construction of the pope to counterbalance the Byzantine Empire. What's interesting is considering where the Holy Roman Empire was located for most of its history - in Germany, a place the Romans never really conquered!
The Ottomans are not considered successors because they had no real connection to the Roman Empire itself except for conquering the lands that were part of the Byzantine Empire, despite what Mehmed called himself.
Other people have provided good answers to your question, but I'd like to add that you should be wary of "traditional" historiography. In the 19th and 20th centuries, European states and nationalist movements created historical narratives that demonstrated the glory and ancientness of their nation. Both France and Germany claimed a link to the Roman empire through Charlemagne / Karl der Große, and for this the "Byzantine" empire had to be put aside as an alien entity. In a somewhat contradictory fashion, both also quasi-deified anti-Roman leaders of whom we know little (Vercingétorix and Hermann), so as to also claim an ancient love of freedom.
This kind of nationalist historiography has been mostly driven out of academic circles and high school textbooks, at least in those two countries, but it is still commonly found in primary school books, "popular history" and public discourse.
Also, this is completely unrelated but "resp." isn't an exact equivalent of "bzw." "Respectively" can be used when drawing a parallel between two things, for instance : "Vercingetorix and Arminius were almost worshipped in, respectively, Republican France and Imperial Germany" and "The Celtic (resp. Germanic) leader defeated the Roman legions at the battle of Gergovia (resp. Teutoburg)."