The replacement of the genitive through the dative has been and still is a highly debated matter in terms of prescriptivism (i.e. what is "correct" and what people should say or avoid). This relates to cases in which a preposition that canonically used to license the genitive has become more and more entrenched with the dative in colloquial speech:
1) Ich rufe an wegen EINES TerminS (Gen.)
2) Ich rufe an wegen EINEM Termin (Dat.)
The case you quote, however, has nothing to to with this phenomenon. In fact,
in your examples we are dealing with two different constructions neither of which
is formal or informal.
In: "Es ist Tradition, dass die Frauen die Krawatten der Männer abschneiden" you are only specifying whose ties the women are cutting.
In: "Es ist Tradition, dass die Frauen den Männern die Krawatten abschneiden, als Symbol der Macht" "den Männern" emphasizes that the possessor is being affected (in this case negatively) by the action at issue. In terms of semantic roles, the dative encodes a "Maleficiary". The possessive pronoun is usually omitted because it can be inferred, but in some dialects it can also be realised: "ihren Krawatten".
Another example of this construction would be:
"Ich wasche meinem Vater das Auto"
In this case, the dative encodes someone who is positively affected by the action,
i.e. a Beneficiary in terms of semantic roles.