When to use the former and when to use the latter? Example:
I followed the girl's scent until it was masked by the chrysanthemums'.
I followed the girl's scent until it was masked by that of the chrysanthemums.
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Careful readers will notice the apostrophe in the word chrysanthemums' and will correctly interpret it is as an ellipsis of the chrysanthemums' scent.
Even less observant readers who don't notice the apostrophe will no doubt correctly interpret the meaning of the sentence in question.
But there are other sentences which may well be misinterpreted by less careful readers. For example:
I followed the girl's dog until it was attacked by the boy's.
They may believe that the dog was attacked by the boys.
If your intention is to avoid any possibility of ambiguity, then it is safer to use that of:
I followed the girl's dog until it was attacked by that of the boy.
Of course, in spoken language (without the apostrophe marker) the possibility for ambiguity is greatly increased.
There is another discussion of this issue here: What's the exact usage of "that of"