I think there's more similarity than difference, however, some of the differences are explainable based on cultural and genetic micro variations.
Culturaly: Language and vocal expression change how one thinks and behaves, as well as influences their vocal development as they age. The tones and vocalizations they produce during laughter are highly influenced by these things. The trigger for laughter is built into the nervous system and structure of the brain, but how it is expressed is highly influenced by other behavior and modeling from those they are exposed to in their immediate social groups. Culture, language, and geography have kept people separated enough that they are not as influenced by those farther outside of their circle of familiarity, thereby leading to natural pockets of difference.
Genetically: There are all kinds of micro differences between races due to adaptations and maladaptations over time. Think skin, hair, and eye color, or physical size, pain tolerance, etc. Some of those genetic variations (and also developmental variations due to geographical resource differences) likely influence the sound of one's voice and speech patterns. For example lung capacity, size of the upper torso, vocal cord development, tongue size and shape, and nasopharyngeal shape and space may all influence the sound of one's laugh.
Again, most of these differences are subtle, but interesting nonetheless. I think it gives interesting character to the individual. You can also hear some patterns of difference in certain general geographic regions where the race is the same, but the subcultures and similar things are different. I can hear this in the US and the UK. I'm sure it's true elsewhere, but my ears are mostly tuned to a handful of traditionally English speaking regions, so I can't comment on many others.