The baking recipes I use specify many kinds of sugar e.g. caster, raw, white, soft etc.
Generally, I always just use raw sugar, mainly for convenience.
Does it really make a difference?
Sugar is often used as a "wet" ingredient in baking. That means it needs to be dissolved in the water in order to prevent too much gluten from being produced (making the result fluffy/flaky, and not chewy). Different sugars hold different amounts of moisture (for example, brown sugar holds more than white) and using sugar with crystals that are too large (or too small) will make the texture come out completely wrong. Too small and your cake will be rubber; too large and your biscuit will fall apart.
Also, if you're using the sugar for creaming (with fat), you generally need to use a coarser sugar. Superfine sugar will dissolve too quickly and won't allow enough air to be incorporated. Confectioner's sugar is good for creaming but has completely different properties from crystal sugar - it's been "processed" and has cornstarch added, so you can't just substitute in equal quantities.
Of course, it goes without saying that the taste is different as well. But even more important than the taste are the solubility characteristics which, as explained above, will have noticeable effects on other parts of the recipe.
It does depend what you're baking; if the sugar is being used purely for flavouring, then you can use whatever sugar you like. Much of the time in baking though, sugar is used for more than just sweetening, and it's important to be aware of that. If the sugar is being creamed or dissolved, don't substitute unless you're sure you know what you're doing.
There's also a difference between brown and white sugar even though they have roughly the same texture/consistency. Cookies will be chewier if you use more brown sugar.
Yes, very much so.
There are many types sugar: icing/confectioners sugar, light brown sugar, dark brown sugar, caster/super fine sugar,Demerara sugar, granulated sugar to name just a few.
Each of these (with the exception of brown sugars and Demerara) can be also found in a 'raw' form (as opposed to 'white') where the molasses (a dark, treacle-like by-product of sugar production) is added back into the sugar resulting in a darker sugar. These raw sugars can be substituted into any baking product specifying for white sugars (as can white sugar visor-versa) resulting in products with a more pronounced 'caramel' flavour. The only thing to note is that raw sugars in a meringue results in a softer meringue.
Icing sugar is generally only used in frostings and icings, particularly in 'quick frostings' and glacé icings. In cake products they give the finished product a drier, almost 'crusty' result. However, they are used in melting moment biscuits.
Caster sugar and granulated sugar can be substituted for each other, although using caster sugar creates better meringues which 'melt in the mouth' better as the finer crystals are more easily dissolved. Saying that, it wouldn't be a disaster at all if granulated sugar was used.
Brown sugars have more moisture in them due to the presence of more molasses but can also be substituted in most uses. Dark brown sugar has more molasses than light brown sugar.
Demerara sugar is often used in crumble toppings and occasionally in some biscuits owing to its coarse texture. However it is not often found in cakes or frostings.