As Fumblefingers mentioned, these are headlines, so normal grammar rules don't necessarily apply. The following explanation describes why it could be grammatically correct to use both singular and plural verb forms with the same word.
Taliban is a plural of the arabic word talib, which means student, because many of the original members of the Taliban organization were students from Islamic schools in the Pashtun region of Afghanistan.
A group of members or the Taliban organization can be, and are, referred to as taliban- students: this usage should be considered plural. The Islamic fundamentalist group is also called Taliban: this usage should be considered singular.
Taliban take key city close to Kabul as U.S. troops arrive to assist evacuations
In this sentence, the word taliban is used to refer to a group of fighters taking a city. The fighters are plural, so the verb is take.
Taliban advances in Afghanistan, U.S. and Britain to evacuate embassies
In this sentence, the organization as a whole is advancing: the organization is singular, so the verb is advances.
This singular/plural duality also occurs in English. The word "fish" can be used to describe one fish, a group of fish (or even a piece of one fish). Both of these sentences are therefore grammatically correct:
The fish swims in the river
The fish swim in the river