If a planet is completely unaspected, the traditional idea is that it's essentially isolated or cut-off from the rest of the chart, such that it has no connections or influence at all. That isn't entirely true, because the planet still has some influence, but it has no way to readily or easily exert or distribute its influence; in other words, it's at a severe disadvantage, and has to try harder to make its influence felt (according to this way of looking at it). In psychological terms, the more aspects a planet has, the more highly integrated it is within the personality, so an unaspected planet isn't integrated into the personality.
However, some modern astrologers have studied unaspected planets, and they've mostly come to the conclusion that being unaspected can actually be a source of strength for that planet – in a way. One way to think of this is that if a planet has to try harder, it may eventually develop its "muscles" and become more powerful. Or if you think of an aspect as a sort of conversation (or interchange) between two planets, you might say that an unaspected planet is being ignored by the other planets, so it may eventually learn to talk much louder to make itself heard.
Of course, it's also true that the planet might just give up on trying to accomplish its aims, or to make itself heard, and go crawl away and hide in the corner, so to speak. It's probably not a good idea to assume that unaspected planets will automatically become strengthened if given enough time, eventually reaching the point where they're very powerful. Nevertheless, an unaspected planet might be overpowering in some way, in the same sort of sense that an isolated personality factor can become overwhelming if it isn't constructively integrated into the personality.
But, there are still many ways in which planets can relate to each other. For example, when a planet is in a given sign, it has a special relationship with the planet which rules that sign – it is "disposited" by that planet. An unaspected planet is still disposited by some planet (although it may be disposited by itself, if it's in one of the signs it rules), and it may (or may not) disposit one or more other planets. This would certainly need to be taken into consideration.