If you do go for a combo, you can almost always improve the sound drastically by buying a set of better speakers. Some of the best combos come without speakers, forcing you to do this. A good combo with replacement speakers will give you very pleasant music. A good component system will normally require a mixture of boxes from different makers to get the best results, so you've got to spend more time listening to things. However, if you listen to your music seriously, then the performance of a component system is the reward for that extra work.
Components are harder to set up and operate.
However, as noted, you can get better sound. You also get more flexibility. If, for example, you decide you want a better CD player, you just replace the CD player. With a combo system, you've got to replace the whole system. If your component tape deck breaks, you can remove it from the system and take it in for repair or replacement.