Consider the following example:
divide the main folder into sub folders (in case of mail)
What I want to know is why do we add "ふり"? Why can't we just use "フォルダをわけて"?
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分ける just means dividing one thing (or several things) into two or more pieces (or groups). 振り分ける means sorting or distributing many things into subgroups, and it is used especially when items arrive over time, as in e-mail filters or HTTP load balancers. In your case, there is only one main folder but there are many mails in it, so you can say メールを(サブ)フォルダに振り分ける or フォルダを2つのサブフォルダに分ける. フォルダを振り分ける doesn't look correct to me because it sounds like there are dozens of random folders to sort.
Regarding this 振る, it means something like "to assign" here. See the third definition here.
フォルダを分ける to classify, to sort, to divide an aggregate into sub-folders when you don't specify a purpose. Probably when a lot of miscellaneous things in one folder, you might feel the necessity to divide the one folder into several folders without specific labels for the moment.
子供と大人に分ける to split one group into the group of children and the group of adults.
大きさによって分ける to classify by the size.
I think "分ける" is focusing on quantifiable things when you classify, split, divide, etc.
フォルダを振りわける to classify, to sort, to divide an aggregate into sub-folders when you have a specific purpose. Such as 生徒を理系と文系に振り分ける to arrange students into science majors and literature majors. They have some reasons for the academic institutions to divide students into sub-categories.
I think 振り分ける is focusing on the qualitative things when you classify, split, divide, etc. such as 生徒を理系と文系に振り分ける is implying classifying the students with their aptitudes.
I do not know what 振り in 振り分ける is.