Free weights (dumbbells and barbells) not only exercise the muscles you’re targeting, but also bring other stabilizing muscles into action. This is the way exercise happens in day-to-day life, and brings overall strength to the area exercised. You can gain greater overall strength with free weights because of this, and your power will likely be more naturally applicable to day-to-day activities. Dumbbells permit this multi-muscle movement to the greatest degree, barbells slightly less, because two hands on a bar (as with a barbell) reduce the planes (angles) upon which the exercise can take place.
Free weights by their nature allow multiple variations of grips and angles and positions, enabling strength building across a wide variety of movement. Lying on your back, facing forward, and lying on your side, for instance, each offer entirely different emphases to a movement. Provided you have an appropriate bench and perhaps a Swiss Ball, there are almost unlimited exercise angles and combinations you can undertake with free weights.
Furthermore, it's often the case that you can move much more quickly between exercises with free weights than you can with machines. For instance, you can change from bench presses with dumbbells to triceps kickbacks in seconds, simply with a change in body position.
Free weights will generally take up less space and cost less. Pin-loaded adjustable dumbbells, such as PowerBlocks, definitely take up the least space of any strength-building equipment.
Strength machines, on the other hand, can sometimes be easier to use. Because they limit the range of movement for the resistance they provide, they effectively guide your movements, where with free weights you need to concentrate on form – that is, doing things the right way. (There is thus a greater degree of injury risk with free weights).
Because they limit movement to specific angles and ranges, strength machines can isolate a particular muscle to be developed, rather than recruiting all of the other muscles around to stabilize it and thus get stronger as well. (There again, there are times when this can be a disadvantage – weak stabilizing muscles, say, in the shoulder, are often injured when larger associated muscles - for instance, chest muscles during a bench press - promote a push that is too much for others, and the others tear - in this case, shoulder muscles) So while the machine approach controls your movement and emphasises a particular muscle, having to concentrate on proper form - say with dumbbells - gives you a better workout, and strengthens those stabilising muscles.
For similar reasons, you'll probably find that if you bench press 100kg on a machine, you'll find your max is 90kg with a barbell and 80kg with dumbbells, for instance. This is natural.
Strength machines tend to built for people of closer-to-average height, so tall or short people can often not get a satisfactory movement “fit”.
So - strength machines help you avoid poor form, can be easier to use, and they make cheating movements difficult. (There's always someone in a gym tossing dumbbells as if he's starting two mowers at once-which is all wrong!) But if you're prepared to put the effort into your form and technique, free weights will develop more complete strength and offer less injury risk.
On balance, this site goes for free weights, and more particularly, dumbbells.