Difference between surge arrester and lightning arrester?

This is mainly a matter of semantics. Historically, the motivation for using arresters was to deal with lightning, so they were called 'lightning arresters'. But as system voltage increased, it became apparent that there could be switching-induced surges that were more damaging than lightning, so today, the preferred term is probably 'surge arrester'.

There is a technical distinction between a 'surge suppressor' and a 'surge arrester' that has to do with the energy dissipation capability built into the device. The device that most people plug computers into is a 'surge suppressor' and has a fairly low energy dissipation capability. It also has a voltage rating that is only slightly greater than the normal system voltage. A typical 'surge arrester' has a far greater energy dissipation capability, and the voltage rating is driven more by the surge voltage withstand capability of the insulation being protected than by the rated system voltage.