By the end of the 7th century BC, there were many problems in Athens. There were economic problems in the polis and there was factional infighting among the aristocracy. It was in this context that first Drakon (usually called Draco in English) and then Solon handed down laws that were meant to quell the disturbances.
It is believed that Draco handed down his laws around 621 BC. We should, however, note that it is not clear exactly what those laws said and there are even those who are not convinced that Draco ever existed. Be that as it may, the laws attributed to Draco tried to set out harsh legal penalties for various offenses. This was important because Athens was still a rather tribal society at that point and people tended to react to crimes by trying to take personal vengeance on those who committed the crimes. Draco attempted to put an end to this by making crimes a matter of law, not of personal vengeance.
About 30 years later, Solon tried to combat very similar problems in a much more liberal way. Instead of trying to enact harsh punishments, Solon attempted to attack the root causes of the conflict. He did things like cancelling all debts that people owed on land. He freed people who had been enslaved because they could not pay their debts. He set up a system that would allow more people to participate in government.
Thus, Draco tried to remedy the problems through harsh punishment while Solon tried to do so by attacking the root causes of the conflicts.