What are the correct ways to pronounce the ב (bet)? Are there variant traditions? Is there an academic/linguistic opinion on how it was most likely pronounced from the time of the giving of the Torah to the writing of the Talmud? How old are the 2 variant pronunciations of with and without a dagesh? What are the earliest references to this distinction? Is there any halachik significance to pronouncing it one way over the other? Is it just considered a hiddur?
It's difficult to place exactly when the difference in pronunciation occurred. The difference is extremely subtle. The original way to pronounce the Veth is called a Voiced Bilabial Fricative. In IPA this is usually transcribed as /β/. American and Israeli Jews tend to pronounce the V as it was pronounced in Europe, as a "Voiced Labiodental Fricative", transcribed as /v/. The later is probably not the original pronunciation. While it is common in European languages, it is extremely rare in the rest of the world.
There also reason to believe that this was not the pronunciation as of the time of Masoretes based on the Niqqud system. The difference between a letter with and without a Dagesh Qal is always whether or not the letter is a "Fricative" or a "Plosive". The Beth is always pronounced as a "Voiced Bilablial Plosive", or a /b/. Without a dagesh, you would expect the same sound except a Fricative, or /β/.
By extension, you can also extend this to Peh and Feh. A Peh is a "Voiceless Bilabial Plosive" or a /p/. Without a dagesh, you would expect a "Voiceless Bilablial Fricative" or a /ɸ/. You do see this in middle eastern dialects. The European Feh is pronounced as a "Voiceless Labiodental Fricative" or an /f/. So the difference is relatively minor. For both letters, it simply goes from a Bilabial place of articulation, to a Labiodental place of articulation. I have pasted a video below which shows the difference. Again, it's subtle.
As for the question of halakhic significant vs. hiddur, I answered something similar in a previous question.
In general from what i've noticed, from a linguistic perspective, the sound is mostly the same from different communities, but how they make that sound differs, and it can cause changes in pronunciation.
Almost everyone pronounces בּ the same, as a b sound, using both front lips.
But the ב tends to have differences in how it's pronounced. Many communities from America, or Spanish speaking countries, make the v sound by placing their lower lip against their upper teeth. However, Jews from Arabic countries pronounce this sound by using their lips only as Arabic does not have a v sound. As a result of this, many Jews from Arabic countries have lost the ability to use the v sound and so they pronounce every ב as a בּ. To see the difference in lip usage, please see this video: