The notes and the sharp/flat pairs you refer to are only the same if you are tuning to 12-tone equal temperament. But they become different pitches if you are tuning to just intonation, Pythagorean intonation or some other temperament. The examples you cite in Wikipedia are there to contrast the distinction between 12-tone equal temperament (where there are only 12 chromatic pitches) on the one hand, and other tuning schemes (where there are more than 12 chromatic pitches, utilizing different intervals) on the other hand.
Modern instruments (since roughly the beginning of the 20th century) almost always use 12-tone equal temperament. But in previous centuries, just intonation and many other schemes were used. There were even keyboard instruments with more than 12 keys/notes per octave in earlier centuries.
Musical temperament and the tuning of scales is a vast topic that comes up in various historical, geographic and cultural contexts. It's a difficult subject that requires doing some elaborate maths and listening very critically to the sonic results in order to be able to appreciate the distinctions.
The short answer to the question is that for a bit more than the last 100 years, almost all musicians in the Western tradition (and almost all contemporary musical instruments and their manufacturers) have "given up" on all methods of tuning and temperament other than 12-tone equal temperament. Other tuning schemes are chiefly used by musicians interested in ancient historical performance practice, or more recent avant-garde experimental music. However, there are still musical traditions outside that of Western civilization (Arabic and Middle-Eastern, Indian, Chinese, African) that continue to use other tuning schemes, scales and temperaments. From the Western standpoint, those other traditions are usually labeled under a large heading called "ethnomusicology".
Wikipedia article on Musical Temperament