Both are measures of carbon fixation rate.
RuBisCO activity specifically refers to the rate at which the enzyme RuBisCO fixes carbon to RuBP, and is measured by isolating the enzyme from tissue samples and, usually, using radiolabelled CO2 to measure how much carbon is fixed in a set amount of time in controlled conditions and with controlled supply of substrate.
Carbon assimilation rate is generally a whole-plant or whole-leaf variable, measured on a larger scale. It is usually measured by measuring leaf sugar carbon before and after a set period, again using radiolabelling. Conditions may or may not be controlled, and it can be measured in the field (unlike RuBisCO activity).
RuBisCO activity might differ from net fixation in the short term in plants which have an intermediate carbon fixation step, such as in CAM or C4 plants, if the net fixation measurement includes the intermediates - the pool of intermediate carbon-containing molecules have technically been fixed but not by RuBisCO. However, over longer time periods this effect will disappear, as ultimately all CO2 is fixed or re-fixed through RuBisCO.