Aristotle considered the earth to be composed of the elements Fire, Air, Water, Earth. Should this system be totally disregarded or does it offer some advantage over modern science's system?
To answer the question directly: yes, it should be completely disregarded except as an intuitive historical artifact. We have a much better understanding of what things are composed of now, and those details are within even a child's capacity to understand (e.g. phases of matter, "metal", "plastic", "glass", etc.).
However, the elements are fun in stories.
These elements actually predate Aristotle going back to pre-Socratic times.
One notices straight-away that three of them correspond to different phases of matter. This is an important distinction and abstraction.
Air=Gas Water=Liquid Earth=Solid
Fire is presumably added to give animation. If all things are made of these elements, then so must be a human being; but what makes him move - fire; think of the phrase "He's all fired up"!
Note, in support of this; that fire is the only element is always active - its flames always flicker; whereas earth, water and air can move as in earthquake, waves and wind; its easy to see we can have them not move - the earth under your foot, the water in a glass, the air in your room. Thus it seems natural that they use fire as an animating principle.
In fact Aristotle added a fifth element quintessence (the name actually means exactly this quint+essence=five+essence). This is because he divided the world into two parts - the earthly and the celestial. Although earthly things are visibly subject to decay and change; the celestial is not - every day the sun rises and stately makes its steady way across the sky. This fifth element is what the celestial sphere is made of. Its uncorruptible, meaning it doesn't undergo change.
As all things on the earth are in different phases of matter - this theory is remarkably accurate. Celestially this isn't true of course. The moon is solid and the sun is plasma.
Presumably Aristotle and his defenders came up with explanations as to how these elements combined and pushed the theory further.
Similar theories arose in India, Tibet and China. The Indians and Tibetans included space as another element, and additionally the Indians included time.
It important as an explanatory step towards modern physics and chemistry. One should also note that development didn't stop there - various authors then proposed Earth, Water or Air as the sole key element; such as Thales who decided on Water. This makes it likely that these elements actually predated Thales too.
This is interesting - as I expect that they collapsed the distinction by seeing for example, ice change to water to steam. Its likely that they had a corresponding idea for condensation & rarefaction which allowed them to change Earth to Water to Air; and back again.
Again this has a correspondance with modern science: This single element corresponds to Matter. Note that this is a correspondance NOT an identity. In other words there are certain resemblences between the two ideas here.
One shoudn't be anachronistic about Ancient Science and that of the Modern Day. Recall that modern science wasn't put-together without any precursors. For example, Newton paid homage to the ancient Greek atomists in his Principia.