There are people that daven Mincha on Erev Shabbos Chanuka prior to lighting the Menorah and there are people that light first and then Daven Mincha. Those that Daven early do it in order to avoid conflict (Tarte DeSasri) as once they light the Menorah it should really be Maariv time. Why is this any different than a regular week with Neiros Shabbos?
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In theory, the split would be:
The main time for lighting Chanuka candles is not the afternoon before, but at night. In cases of necessity, we allow the latter part of the afternoon to "start" the next evening.
On Shabbos, since we pasken like the Ramban and not the Baal Halachos Gedolos or Ran, lighting candles is not the act of accepting shabbos, but a mitzva that we do on Friday afternoon so that we have light on shabbos. Lighting candles on Friday afternoon does not preclude us from saying the afternoon prayers. (See Ran Shabbos 10a in the Ran)
The idea of tartei d'sasrei for chanukah is strange. The timing for ner chanukah is merely an application of pirsumei nissa (dark with people around). [This is certainly true for many who use the timing in the gemara (from shkia until the pedestrians cease) only for the length of time the candles must burn and light in the house later.] Hypothetically, if G-d would have made the evening light and the day dark, we may still be required to daven mincha at 3:00 PM in the dark, but chanuka lighting would not be at 5:00 PM when it gets light out, but in the morning when it's dark. So the fact that someone lit Chanukah candles should not be an indicator of the time for maariv. (Is this making sense? I would not be offende if anyone can edit this to make it clearer.)