Strengths:
Chain mail was relatively easy to make, One simply only had to heat the metal hot enough to be able to bend, and to hammer the links together.
Chain mail was typically lighter than plate armor, and allowed for more free range of movement and complete protection.
Chainmail is extremely strong against slicing motions, such as a sword brought down on a vertical trail, or against axes with the same motion.
Weaknesses:
Chainmail simply did not have the comparative strengths of a full plate armor. Metal that has been reforged and recast and molded do not retain the same quality as one with less tempering.
Chainmail, depending on the time period as well as how much people worked on a particular chainmail, would take awhile to make, as individual links would have to be put together. A chainmail with a experienced team would take at least 500 man hours of work.
Chainmail was particularly ineffective against jabs, particularly from spears or specialized arrows, which could find it's way through the holes. The same problems that plague plate armors also address chainmail, in which blunt weapons such as maces that particularly lean towards the usage of power and strength, typically could cause traumatic injuries.
Note the pictures.
Picture 1 is of chainmail, and you can see the individual links that are used to make the over all chainmail. Putting all of the individual links together would take large amount of time and effort.
Picture 2 is a mixture of plate and chainmail. Plate armor is used to protect broad areas that typically do not see much movement, as well as protection of a man's most important parts (chest, heart). The chainmail is used in areas that will see movement, such as areas near the armpit, as well as the neck.
Picture 3 is full plate armor when armorers found how to create larger pieces of plate armor that could fit together. It was used near the Late Middle Ages, and saw continued usage in Europe until the introduction of firearms.