1960s and 1970s
In the early days anime artists were inspired by western animation like that coming out of the Disney Studios. Disney made animation for movies popular, so the large cartoon eyes and other features were copied by the Japanese. Though cartoony, you can still see that anime was focused on being more real or naturalistic.
Anime continued to move away from the more cartoony western animation, but began experimenting with stylization itself. In part, this was a response to a changing technological climate - previously, all anime were made for movie theaters as shorts or serials, and as the transition to TV anime approached, a way to make the animation faster and less expensive was needed.
1980s
As the 1980s progressed, the anime made for movies retained the more naturalistic and realistic style of the 1970s, while TV anime began getting progressively more stylized. Hair began being colored unrealistic colors, and hair styles got progressively more extreme, as did the features of the characters.
1990s
The early and mid 1990s saw an explosion of stylistic designs and changes with bright contrasting colors and designs, rainbow hair, very sharp features and cell-shading, and giant eyes. The crazier the hair, the better. New technologies were making it easier to produce and color anime, so designers went crazy experimenting. Emotion was king over representational form. The shape of the eyes, style of the hair, and color of the hair were all developed into shorthand to describe a character's personality.
Late 1990s, Early 2000s
In the late 1990s and early 2000s we started to see a definitive "split" in the anime art world. Some anime continued to push the stylization even farther, almost back to a purely cartoon aesthetic, emphasizing the style and cuteness over everything else. On the other hand, due to computer coloring and animating, the production cost of anime had fallen sufficiently that it started to become possible to emulate the old movie style anime of the 1970s and 1980s on TV, and many designers embraced a return to much more realistic forms and colors. While the stylized side of the anime world had almost done away with noses by this point, on this side of the design gap, noses suddenly became the most realistic they had been in over 20 years. Eyes shrank, and neutral and realistic color palettes appeared for perhaps the first time ever.
Today
The divide in styles still remains, but the realistic design side seems to be winning the battle handily. Even in stylistic anime, natural hair colors are the rule rather the exception. Hyper realistic styles are praised and lauded, and even anime that is stylistic pays close attention to the forms of the body and is rigidly realistic in everything but the face. Gone are the hyper shiny hair highlights of the 1990s.
Others have pointed out how the harsh contrasting cell shaded style has given way to the soft shaded forms of modern anime. Part of this is cost. Soft shading used to be very expensive, and unusable for animation. Now, with computers, it is much easier and cheaper. While the cell shading has a tendency to make characters look flat, the softer shading gives them more of a three dimensional presence.
Fewer anime artists are using artistic shorthand now as well. The bodies of anime characters in the 1990s and earlier were very stylized, a long of long or sharp lines. Anime like Cowboy Bebop and movies like Perfect Blue started emphasizing a wider range of body types, and modern anime draws realistic bodies for the most part, even down to the folds and wrinkles in a hand, or showing proper muscle shading and shape in the arms, chest, and legs. The face is really the only thing that is changed from a realistic figure. Unlike the misshapen heads of the past, current anime heads are shaped like there is a real skull underneath, and the only differences come in on the jaw (slightly more sharpened chin or jawline on females), the nose, and eyes.
I'd say one difference (and I don't have any proof of this) is that it seems a lot more anime artists are coming from traditional art backgrounds rather than cartooning backgrounds. They know how to realistically depict the human body, and each change or stylization you see in modern anime is usually very deliberate and controlled for certain effect. I imagine the internet may be pushing things in this direction as well, since anime artists are facing immense competition. Drawing the simplified and stylized anime of the 1990s isn't going to cut it when half of DeviantArt can copy that style in a hot minute. Instead, anime artists are pushing forward more detailed, elaborate, and realistic styles with personal changes, because it is both inspiring to look at and impossible to copy without real artistic training and talent.
Now, as for a timeless style?
Look at my examples. Studio Ghibli has had a consistent style for decades, and it continues to work because it is emulating reality for the most part. The thing about the myriad of anime styles in the 1980s, 1990s, and even now, is that the more stylized something is, the more dated it will look later, just like clothing. Styles, by their nature, go out of style. In my opinion, emulating a natural or realistic style is the best way to go. Even some of that stuff from the 1970s looks pretty good today because they were going for a more realistic look, while the bright hair, super highlights, squashed heads, and insane hair of the late 1980s and 1990s looks very dated now, just like those bell-bottoms and pastel suits at the local thrift store.
And if the history of ALL art is any indication . . . everything always goes back to realism at some point. Art will shoot off into wild tangents of style for a while, but it always comes back to realism. Emulating artists like
Range Murata is probably a good starting point. Then carefully add in the stylization you want for the effect you want.