The number one advantage of using a Leica is the glass. The cost of lenses, particularly new or recent makes the cost of the M9 body seem cheap. Of course, if one fashions himself (or herself) to be HCB-21st century style, an M9 and a 50mm Summicron would be enough. As far as the comparison to an SLR in general and a digital SLR in particular, here are the things to consider.
Any modern SLR uses autofocus, which is helpful in general and in tracking moving subjects in particular. The M9 is a manual focus camera and it is a rangefinder. If you have never used a rangefinder you may find it difficult to use. There is a double-image that you align so that there is just one image. For many, the image you see is tiny; the magnification can seem poor (the objects are really small). However, the great advantage versus an 100% viewfinder (whether SLR or LCD screen) is that you can see beyond the image that will be taken. Thus you can snap at the decisive moment of action and freeze the subject at the critical juncture.
Of course, if you couldn't focus, you have the decisive moment rendered but poorly. The M9 has some quirks. It has an LCD screen that is low-res for its price bracket. It is slow to use. You have to remove the entire bottom to get to the card or battery (this is copying the classic M body for no apparent reason). It does not have an anti-alias filter, thus offering the potential for much sharper native images (most sensors require sharpening to deal with the AA filter's effects; look at a RAW image and you will see this immediately). The tradeoff is the possibility of moire for things with stripes or repeating line patterns.
An M9 is smaller but is not much lighter; all Leica gear is built to last forever and made mostly of metal. The lenses are to die for. If you get an M and like it you may end up jumping off the upgrade chain...and even get M film bodies as well as start a collection of Leica gear that can quickly approach or exceed the time and money spent on audio (since you may wish to expand your lens collection, travel so you can take photographs etc.).
The nice thing about the M9 and things like it is that they are smaller, less imposing, esp. compared with an SLR with a massive zoom lens with a 77m front element and hood and thus people are less likely to notice that you are shooting.
My father has an M9, but at his advanced age the focusing is difficult. I like it and may acquire it from him. Otherwise, I use my manual, film M7 and MP, R8 or if digital, things which remind me of the Leica experience, lately the Ricoh GXR.
I would consider the Leica X1 after they offer the firmware update to see if the focus speed improves. Apples to apples, the Ricoh GXR won my vote - same sensor, versatility of lenses, current SOTA LCD screen and excellent fairly fast prime lenses (28 and 50mm equivalent f2.5).