In this case, the em dash is taking the place of a semicolon, possibly providing slightly more emphasis on the break:
The periodic table proposed by Mendeleev was the culmination of many observations and discoveries; but, like every culmination, this end was also a beginning.
The periodic table proposed by Mendeleev was the culmination of many observations and discoveries — but, like every culmination, this end was also a beginning.
In general, the em dash is fairly versatile. It can take the place of commas, parentheses, colons, and semicolons. The trick is not overusing them. There's a reason that the standard punctuation marks exist, and in most cases, it's best to use them.
The em dash can be used to add an occasional, and perhaps more informal, flair to your writing.