Although it must still be activated by the phagocytic cells of the innate immune system, the huge new advantage of adaptive immunity was a cluster of three related attributes: specificity, clonal expansion and memory.
Let's say a particular bug (e.g. Streptococcus pneumoniae) is rather common and often deadly. With only the innate immune system functioning, you're no better off the second time, or the third time, or the tenth time, than you were the first time. And of course, there may not be a tenth time - sooner or later, if you get enough infections from this bug, one of them will kill you.
What would really help in this scenario is if your body could learn from its first exposure and ready specific defenses for any further assaults it may have to deal with. This is, in a nutshell, the chief aim of the adaptive immune system (although there are others). It does this by creating an unbelievably huge collection of receptors for almost any possible pathogen. (This large number is necessary because new ones are evolving all the time.) Typically each cell of the adaptive immune system contains just one type of receptor for one type of antigen. Should an infective agent breach your body's defenses, one of the trillions of trillions of receptors will be just right at recognising it, and if the receptor binds to the pathogen, it gets handed over to the innate immunity's phagocytes, which kill it.
So far, not much too different. However, the adaptive immune system's cell now divides exponentially to dramatically boost the number of cells that are specific for that pathogen. This greatly increases the body's ability to rid itself of the organism. Furthermore, a subset of these immune cells is kept in circulation long after the infection has been cleaned up. In this manner, the immune system has a kind of databank of past infections, and is ready to respond quickly and forcefully should any one of them recur. Thus the three characteristics of the adaptive immune system (specificity for a particular antigen, clonal expansion of these specific cells if the appropriate antigen arrives, and memory of past infections) offer another set of strings to its bow.
The adaptive immune system's cells are all lymphocytes. T lymphocytes engage only in direct cell-to-cell interactions, whereas B lymphocytes secrete soluble molecules called antibodies into the blood stream. These act like distress flares; once they attach to their antigen, they set into motion many of the other aspects of the body's immune system (including phagocytosis and complement), thus clearing the antigen.
Finally, the adaptive immune system is slower to respond than the innate one, especially initially. This is principally because it is more complicated and begins with a much smaller pool of activated cells (perhaps only a single lymphocyte!) from which things must progress. However, although it doesn't ever quite catch up to the innate immune system's speed, the adaptive immune system is much quicker on re-exposure to a particular antigen, thanks to a waiting pool of memory cells the second time around.
OK, now the most efficient way of answering the original question is, sadly, in table format, but the above should make this all a little easier to memorise. The one below is my adaptation of a table found in a journal article entitled "The Immune System in Critical Illness".
|Characteristic||Innate Immune System||Adaptive Immune System|
|Specificity||Nonspecific pattern recognition||Specific antigens|
|Activation||Rapid||Delayed, especially intially|
|Cells involved||Macrophages, neutrophils, NK cells||T and B lymphocytes|
|Molecules involved||Complement, acute phase reagents||Antibodies|
|Memory||None; all preprogrammed||Key feature|