You will not observe any difference. Benzyl bromide is as horribly lacrymating as benzyl chloride. The surprising difference is, that both compounds have an official ("harmonised") EU classification where benzyl chloride is carcinogenic but the bromide is not even suspected to be carcinogenic.
Therefore it is a good idea to replace the chloride by the bromide. The bromide is even more reactive than the chloride but therefore it also decomposes more rapidly. But since also the chloride is often used only "freshly distilled" this is no disadvantage. On the contrary you may make use of the fact that the bromide has a melting point of -1°C and therefore will become a stable solid in the freezer.
If a very pure quality is needed, a simple distillation is not successful for both compounds. The bromide is purified by washing with concentrated sulfuric acid, water, 10% sodium carbonate or hydrogen carbonate solution, then again with water and finally dried over calcium chloride or magnesium chloride. The vacuum distillation is performed with exclusion of light. (W.L.F. Armarego, D.D. Perrin, Purification of Laboratory Chemicals 4th Edition 1996, Butterworth-Heinemann)
It has to be accepted that a molar equivalent of the bromide is 10 times more expensive than the chloride. But since modern laboratory experiments are mostly performed as small scale reactions this should be not much of a problem.
Benzyl bromide can be prepared in a student's lab course by radicalic bromination of toluene. If the yield is similar to values described in literature the overall costs for this experiment are similar to the cost of benzyl bromide purchaised from a distributer. Even if the yields are lower this could be still a win-win-situation since the students anyway have to prepare compounds in their lab courses and a useful compound of course is better than a compound prepared for the trash bin.
Since the benzyl bromide is highly lachrymating the students need a good advice, if they have to prepare or use this compound. A German example is found at FU-Berlin (Experiment 2-3).