Why does an OSM map on my Garmin look different to the web site?

There must be over a hundred different extracts of OSM data for Garmin devices. Each may choose to select specific subsets of OSM data, and to style it in the way that the author of the extract chooses. Similarly the mapnik render on the main OSM site is just one of many different ways in which OSM data can be rendered (OpenCycleMap and Osmarenders are also available on the little (+) tab on the right-hand side of the screen to demonstrate this difference).

So first of all, OpenStreetMap is about the data, not specifically how it appears. There are no requirements for how any consuming application as to which data it should use or how it should present that data. This is core aspect of the whole idea.

Secondly, as you do not say which Garmin extract or which Garmin device you are using, it is impossible to identify where your problem lies. I use several extracts which all show the type of information you mention on my eTrexs and in MapSource. More information is available at OSM on Garmin/Download, and details of how maps are created at Mkgmap on the OSM wiki.

The maps on the website are PNGs rendered from Openstreetmap data with a renderer like Mapnik or Osmarender. You can select between several styles by using the white + on blue ground in the upper righthand side.

The file you load on your garmin unit is a vector representation of the Openstreetmap data in Garmin's own file format. It is not rendered with one of the standard renderers and thus does not look like the map on the website. Some downloadable garmin data include a "type" file that tells the unit how to render certain things, e.g. forests. Depending on the typefile the map on the garmin unit can look very different. I don't know any type file that is designed to conform with the look of the rendered tiles though.

Just to add a bit more to what SK53 said - the default OSM style is called "Mapnik" (after the software used to create it). Various people have tried to create "Mapnik-like" maps for Garmin devices - there are some listed on the wiki page that SK53 linked to. However, the options on the screen of a small device are limited when compared to what you can usefully display on a large PC screen.

If you want to, you can change what data is displayed - you can create your own maps and (more difficult) your own map style by creating a "TYP" file that defines it. Before doing that I'd suggest trying searching through the mkgmap development list for things that you might be interested in (like this for "mapnik").