Originalmente, "diferenciales" y "derivados" estaban íntimamente conectadas con la derivada se define como la proporción de la diferencial de la función por la diferencial de la variable (véase mi discusión anterior sobre la la notación de Leibniz para la derivada). Diferencias eran simplemente "infinitesimal" en whatever, y el derivado de la $y$ with respect to $x$ was the ratio of the infinitesimal change in $y$ relative to the infinitesimal change in $x$.
For integrals, "differentials" came in because, in Leibnitz's way of thinking about them, integrals were the sums of infinitely many infinitesimally thin rectangles that lay below the graph of the function. Each rectangle would have height $y$ and base $dx$ (the infinitesimal change in $x$), so the area of the rectangle would be $y\,dx$ (height times base), and we would add them all up as $S\; y\,dx$ to get the total area (the integral sign was originally an elongated $S$, for "summa", or sum).
Infinitesimals, however, cause all sorts of headaches and problems. A lot of the reasoning about infinitesimals was, well, let's say not entirely rigorous (or logical); some differentials were dismissed as "utterly inconsequential", while others were taken into account. For example, the product rule would be argued by saying that the change in $fg$ está dado por
$$(f+df)(g+dg) -fg = fdg + gdf + df\,dg,$$
y, a continuación, ignorando $df\,dg$ as inconsequential, since it was made up of the product of two infinitesimals; but if infinitesimals that are really small can be ignored, why do we not ignore the infinitesimal change $dg$ in the first factor? Well, you can wave your hands a lot of huff and puff, but in the end the argument essentially broke down into nonsense, or the problem was ignored because things worked out regardless (most of the time, anyway).
Anyway, there was a need of a more solid understanding of just what derivatives and differentials actually are so that we can really reason about them; that's where limits came in. Derivatives are no longer ratios, instead they are limits. Integrals are no longer infinite sums of infinitesimally thin rectangles, now they are limits of Riemann sums (each of which is finite and there are no infinitesimals around), etc.
The notation is left over, though, because it is very useful notation and is very suggestive. In the integral case, for instance, the "dx" is no longer really a quantity or function being multiplied: it's best to think of it as the "closing parenthesis" that goes with the "opening parenthesis" of the integral (that is, you are integrating whatever is between the $\int$ and the $dx$, just like when you have (84+3)$, you are multiplying by $ whatever is between the $($ and the $)$ ). But it is very useful, because for example it helps you keep track of what changes need to be made when you do a change of variable. One can justify the change of variable without appealing at all to "differentials" (whatever they may be), but the notation just leads you through the necessary changes, so we treat them as if they were actual functions being multiplied by the integrand because they help keep us on the right track and keep us honest.
But here is an ill-kept secret: we mathematicians tend to be lazy. If we've already come up with a valid argument for situation A, we don't want to have to come up with a new valid argument for situation B if we can just explain how to get from B to A, even if solving B directly would be easier than solving A (old joke: a mathematician and an engineer are subjects of a psychology experiment; first they are shown into a room where there is an empty bucket, a trashcan, and a faucet. The trashcan is on fire. Each of them first fills the bucket with water from the faucet, then dumps it on the trashcan and extinguishes the flames. Then the engineer is shown to another room, where there is again a faucet, a trashcan on fire, and a bucket, but this time the bucket is already filled with water; the engineer takes the bucket, empties it on the trashcan and puts out the fire. The mathematican, later, comes in, sees the situation, takes the bucket, and empties it on the floor, and then says "which reduces it to a previously solved problem.")
Where were we? Ah, yes. Having to translate all those informal manipulations that work so well and treat $dx$ and $dy$ as objects in and of themselves, into formal justifications that don't treat them that way is a real pain. It can be done, but it's a real pain. Instead, we want to come up with a way of justifying all those manipulations that will be valid always. One way of doing it is by actually giving them a meaning in terms of the new notions of derivatives. And that is what is done.
Basically, we want the "differential" of $y$ to be the infinitesimal change in $y$; this change will be closely approximated to the change along the tangent to $y$; the tangent has slope $y'(a)$. But because we don't have infinitesimals, we have to say how much we've changed the argument. So we define "the differential in $y$ at $a$ when $x$ changes by $\Delta x$", $d(y,\Delta x)(a)$, as $d(y,\Delta x)(a) = y'(a)\Delta x$. This is exactly the change along the tangent, rather than along the graph of the function. If you take the limit of $d(y,\Delta x)$ over $\Delta x$ as $\Delta x\to 0$, you just get $y'$. But we tend to think of the limit of $\Delta x\to 0$ as being $dx$, so abuse of notation leads to "$dy = \frac{dy}{dx}\,dx$"; this is suggestive, but not quite true literally; instead, one then can show that arguments that treat differentials as functions tend to give the right answer under mild assumptions. Note that under this definition, you get $d(x,\Delta x) = 1\Delta x$, leading to $dx = dx$.
Also, notice an interesting reversal: originally, differentials came first, and they were used to define the derivative as a ratio. Today, derivatives come first (defined as limits), and differentials are defined in terms of the derivatives.
What is the practical difference, though? You'll probably be disappointed to hear "not much". Except one thing: when your functions represent actual quantities, rather than just formal manipulation of symbols, the derivative and the differential measure different things. The derivative measures a rate of change, while the differential measures the change itself.
So the units of measurement are different: for example, if $y$ is distance and $x$ is time, then $\frac{dy}{dx}$ is measured in distance over time, i.e., velocity. But the differential $dy$ se mide en unidades de distancia, porque representa el cambio en la distancia (y la diferencia/entre dos distancias es todavía una distancia, no es una velocidad más).
¿Por qué es útil la distinción? Porque a veces usted quiere saber cómo algo está cambiando, y a veces usted quiere saber cuánto algo cambiado. Todo es lindo y bueno para saber la tasa de inflación (cambio en los precios a lo largo del tiempo), pero puede que a veces quieren saber cuánto más el pan es ahora (en lugar de la tasa en la que el precio está cambiando). Y porque ser capaz de manipular derivados, como si fueran los cocientes puede ser muy útil cuando se trata de integrales, ecuaciones diferenciales, etc, y las diferencias nos dan una manera de asegurarse de que estas manipulaciones no llevarnos por el camino equivocado (como a veces lo hizo en los días de infinitesimals).
No estoy seguro de si eso responde a su pregunta, o al menos da una indicación de que las respuestas están. Espero que sí. Añadido. Veo Qiaochu ha señalado que la distinción se vuelve mucho más clara una vez que usted vaya a mayores dimensiones/variables de cálculo, de modo que las anteriores puede ser un desperdicio. Todavía...
Añadido. Como Qiaochu puntos (y que he mencionado de pasada en otros lugares), no son maneras en que uno puede dar formal de las definiciones y los significados de infinitesimals, en cuyo caso se puede definir diferenciales como "infinitesimal cambios" o "cambios infinitesimales de las diferencias"; y, a continuación, utilizar para definir los instrumentos financieros derivados integrales igual que Leibniz hizo. El ejemplo estándar de ser capaz de hacer esto es Robinson no estándar de análisis O si uno está dispuesto a renunciar a buscar en todos los tipos de funciones y sólo en algunos de tipo restringido de funciones, entonces usted también puede dar infinitesimals, diferenciales, y los derivados de la sustancia/significado que está mucho más cerca de su concepción original.
