Does anybody know why they don't use the characters in this image anymore?
sometimes, languages change over time to adapt to the way they are used, and languages almost always lean towards what's easier and more natural to pronounce.
The bilabial approximate "w" can easily be elided (unspoken) depending on the phonetic context. Say "wa, wi, wu, we, wo" over and over, faster and faster, then do the same thing for "a, i, u, e, o." At a certain point, it will be difficult to distinguish the sounds with and without the "w."
Since Japanese syllables almost always end in a vowel, the "w" in these kana would often be sandwiched between vowels, making them harder to speak, and also harder to hear at times. The only one that stands out amongst these is "wa," which is why it still exists in the Japanese syllabary ("wo" is still kicking around but it's almost always used to distinguish it as a particle, and that is not even pronounced with the [w]")
In English, misspellings caused by mishearing can make their way into the lexicon and eventually become accepted and even "proper" spellings. We can see some of these changes happening in Japanese as well, as "atashi" has become an acceptable feminized form of "watashi," with the [w] elided. Without a lot of minimal pairs (two distinct words that are identical except for one sound), the need to clearly enunciate the sound is diminished, which can lead to the sound changing or disappearing altogether in some dialects. This is one mechanism for how accents develop, and why the way we speak is often a reflection of where we're from, how well-off our family was, and the "quality" of our upbringing and education.