Both are devices to draw a graph of voltage against time.
Logic analyzers are optimized to record digital signals. Oscilloscopes are optimized to record analog signals.
Internally, an oscilloscope, no matter what technology it uses, will show a trace displaced by an amount proportional to the analog signal voltage fed into it.
Logic analyzers do not need to see anything other than 'high' or 'low' at their inputs so that is all they record. They often have other features like being able to start recording when a particular pattern on the inputs is recognized.
The amount of memory depends on several things but primarily on the amount of information you want to store and how long is to be spent capturing it. As these increase, so does the memory requirement. Remember that to capture data accurately, you need to 'snapshot' it, digitize it and store it quickly enough that the positions of rising and falling edges (the pulse length) is not seen to be too far from real position.
Brian.