Synchronization of two time-delayed chemically coupled neurons with burst-spiking states is studied. Different from the previous study by N. Buric et al. (Phys. Rev. E 78, 036211 (2008)), it is found that exactly synchronous burst-spiking dynamics can occur for small coupling strengths and time delays. The results are confirmed by common time delays and non-equal time delays. When common noise is added to the two neurons, synchronization is enhanced as noise strength is increased. But the results are different for larger time delay and smaller time delay. When noises are correlated, it is found that only strong noises with large correlation coefficient can induce exact synchronization. Even one percent of independent noises can influence synchronization much.