Don't they basically serve the same purpose? Why didn't they just build larger city walls to protect their cities instead of a castle? Why didn't Romans build any castles?
Because castles are not meant to protect cities (which were not that common in the first place) they are military installations meant to project force in a region. A castle is a military base essentially where a lord can project his force to the surrounding area while being safe from raiding or attack.
It's part of why war slowed down so rapidly. You can't force a decisive engagement so war is fought by raiding and siege warfare. You can't take land without taking the castles as if you tried to ignore or bypass them they can assault you with impunity and retreat to their holes. The 80 and 100 years war lasted so long with so few decisive engagements for reasons like this in part.
As time went on they would be placed on geographically strategic places. In passes and on river crossings and mountain ridges. If you wanted to enter or exit a region you had to go through the castle.
So basically the location of a city didn't necessarily align with the most strategic position to put a fort as they were separate entities for separate purposes. Sometimes those interests aligned or sometimes cities were economically vital and were protected equally as well.
As for the Roman question I don't know anything about that!
Follow up to Elos_'s answer. The Romans certainly did build fortifications and strongholds throughout their territories for the same reasons as castles, to include holding key geographical features and projection of force. The former Roman Empire is littered with the ruins of Roman Military fortifications. However, the strength of the Roman Army was not in it's defensive capabilities, but rather it's capabilities as a field army. The organization and tactics employed by the Legions gave them significant advantages when used against an enemy on an open battlefield. A prime illustration of the open field capabilities of the Romans can be seen in The Battle of Watling Street. Rather than retreating to fortifications, the Romans met a numerically superior force on the open battlefield. With careful selection of terrain and implementation of tactics the Romans crushed the opposing forces.