First and foremost, Buddha was not a vegetarian. Basically, he ate whatever decent food was offered to him, just so long it's not poisonous and not an animal that was deliberately killed for him. Yes, Buddha ate meat; he was not a vegetarian.
However, during the period that Buddha lived in India, it was, and still is, a very strong custom for Indians not to eat beef. The reason was because the ox tilled the lands for the farmers and was an utmost valuable contributor to the livelihood and economy of the people. Hence, most folks then regarded the bovine creature as part of the family, conferring on it a status close to that of a precious family member.
Being the wise and ultimately sensitive person that he was, the Gautama Buddha abstained from eating beef too so as not to upset the general social customs then.
Then Buddhism travelled to China via the Silk Route starting from around the 1st century AD. It was through a natural evolution that Buddhism took on certain practices from Confucianism and Taoism while maintaining its core teaching of salvation through the realisation of non-self.
In Chinese culture, there were numerous myths that made its way into Taoism, especially those involving deities and gods. It so happens that there was an interesting story originating from Taoism mythlore that explains why many Chinese do not eat beef. It goes something like this:
A few thousand years ago, there was a huge famine in the land of the Central Kingdom*. The common folk suffered from bad crops and an incredible shortage of staple foods.
The Heavenly Emperor, upon seeing the destitute commoners, decided to send the Horned Ox Deity to convey a message to them and to shine a ray of hope on their poor stomachs. Hence, the Horned Ox Deity lumbered down to earth and made a proclamation on behalf of the Heavenly Emperor:
"Under the auspices and power of the all-mighty Heavenly Emperor, it is hereby proclaimed that his Celestial Majesty will provide at least three meals a day for everyone!"
Everyone jumped up for joy and raised their hands to heaven, thanking the Heavenly Emperor. Many touched their heads and lips to the ground in their jubilation. Riding on a wave of gratitude and thanks from the humans, the Horned Ox Deity flew back and reported the glad tidings to the Heavenly Emperor.
Imagine his surprise when the Heavenly Emperor gave him a tongue lashing upon his return!
"You fool! I said that I'll provide at least one meal every three days for everyone. Not three meals every one day!" fumed the Celestial All-Mighty. "For that, you are banished to earth and shall slog for eternity in order to provide the humans with three meals every day!"
Hence, it came about that the ox had to plough the fields to provide people with proper meals thrice a day. As a result, the Chinese looked upon the ox as a deity and refused to eat its meat.
As Buddhism spread in China and absorbed certain social customs of Taoism, practitioners of Buddhism also adopted the practice of not eating beef, which actually originated from Taoism.
In reality, Buddha did not specify the practice of not eating beef over other types of meat. So, not eating beef certainly does not make one a better Buddhist. If you really want to be compassionate to all beings, how about abstaining from all types of meat and not just beef?
*中国 (China in Chinese characters) is often translated as the Middle Kingdom because 中 literally means 'the middle'. However, my opinion is that the ancients had meant 中 as 'the centre', another less oft-used meaning of 中.