Drawing and watercolor papers are made from cotton pulp, not wood pulp. Even though most of the fiber is 'leftovers' from textile manufacture, it is still a more expensive commodity than wood pulp. Art papers made from wood pulp are treated to neutralize the acids in the wood, which makes even these cheaper papers more costly than plain office paper. Watercolor paper also has more 'size' it it. Size is the name for the glue that holds the paper together, and coats it so that the paint doesn't sink in.
Weight or Thickness
Art paper is thicker than regular sketching paper. Some printmaking papers are as heavy as 600gsm, but your standard heavyweight watercolor paper is still a hefty 300 gsm (grams per square meter) and lighter stuff is about 180 - 220 gsm. Compare this to office paper which is about 80 GSM while a slighly heavier paper (not quite card - the sort you might use for printing an invitation on) is about 110 GSM.
In the case of watercolor paper, the paper has to be thick so that it doesn't buckle and warp as you paint. Thinner paper is cheaper, but you need to stretch it before you use it. Pastel drawings also need a robust paper to reduce movement movement and provide enough tooth to hold the layers of pigment.
Surface
Machine Papermaking runs a continuous sheet through a system of rollers. The directional pull used in this method creates a 'grain' which means the paper will tend to warp and stretch more in that direction. This can be a problem for printmakers, who need a stable paper which won't distort, so that their prints register properly (the colors overlay without mismatched edges). Handmade paper doesn't have this directional grain, and has a 'deckle' (natural rough finish) on all four edges.
Some art papers are specially treated to create a good drawing or painting surface. This may include 'calendering' or hot-rolling, to create a smooth, glossy surface; cold-press embossing to create a rough surface; and even the application of fibers to the surface, to create a velours surface for pastel drawing.
Cutting the Cost
Sometimes you can make do with using cheap paper. I do a lot of sketching with office (bond) paper - it has a nice, consistent, smooth white surface and is very cheap. But if you want your drawings to last, or want to use a paper which will allow you to build up strong layers of graphite, sometimes you have no option but to choose a good art paper.
For some purposes you might choose an acid-free wood pulp paper, or a 50% cotton - 50% woodpulp watercolor paper. Paper varies a lot and you'll want to try several to find one that suits your individual needs. I recently bought what looked like a nice paper that turned out to be horrible - it had hardly any size in it so that the watercolours sank right in - it was impossible to do an even wash. There are also some brands of expensive paper that are too heavily pressed and sized for my liking - mediums just don't stick to their surface.
For ink and watercolor,it is often cheaper way to buy watercolor paper is to buy in large sheets than in readymade pads. I usually cut them in half, then cut one in half again, then another - that gives me two small pieces, one medium and one large. I re-use paper by using the backs of bad drawings to work on or to do excercises on.
If you do a lot of drawing, once you've found the paper you like the best, you should consider buying it in bulk. You can make considerable savings buying it by the pack rather than by the sheet.
Last of all, when drawing with good paper, relax and enjoy it! Don't worry about the cost, or the risk of wasting the paper - that worry will show up in your drawing! Keep some scrap paper handy - including scraps of your good drawing paper - to test out techniques before you use them. Warm up with some rough sketches first, so that you are ready to get the most out of that beautiful paper.
