While Galway offers street entertainment, Cork hosts the best music concerts. For a fantastic dining experience, Cork beats Galway. However, if you want to enjoy the city with minimal movement, go to Galway. Whether you should go to Cork or Galway depends on your interests and what you want to experience.
Galway and Cork are two cities that people often consider as alternatives to visiting Dublin. Galway is a much smaller city than Cork and has the “charm of a small town”. If you want to experience authentic Irish culture, Galway is your choice. Cork is much larger, but it still feels like a smaller city.
By comparing the travel costs between Galway and Cork, we can see that Galway is more expensive. However, the two cities are actually relatively comparable in price, since the difference is somewhat minimal. In general, this means that you can travel with the same travel style and level of luxury in every location. Since both cities are in Europe, it's no surprise that their costs are relatively close, as many destinations have somewhat similar travel prices in general.
This is too much ground to cover with the time you have. With only six nights, you actually only have five days, and it seems that the first one is in Dublin and you have to go back there to fly home. If you were to drive directly to Dingle from Dublin, it would still take a few hours, so you won't have a full day when you get there. Then you'll want a full day to explore the area, so you'll have to spend at least two nights there to enjoy a full day of sightseeing.
Then, the journey from Dingle to Galway also takes a few hours, in addition to the stop at the cliffs and perhaps somewhere else along the way it will last another day. I think that at this point you will only have one full day left until you arrive in Galway. My suggestion would be to focus on the east or west coast, or just the upper half from east to west. Either way, I would skip Cork.
If you're not interested in spending a lot of time in Dublin besides St. Patrick's Day, I suggest you stop visiting the southern part of your trip. You'll see a lot more if you limit your focus. From Dublin you can go to Newgrange, then head to Galway and stop at Clonmacnoise and other places along the way.
Once in Galway, you can easily see the Cliffs of Moher, Burren, Connemara or even take a day trip to Inishmore. But if you want to drive to Dingle and back, you'll spend a lot more time in the car and, basically, you'll have a blurry, green ride out the car window. Totally biased, since I have family nearby, but for me it's Galway. Beautiful city, great base for day trips, good restaurants, etc.
Cork is also a lovely place. I hope that no matter what city you end up visiting, that you have a WONDERFUL time. Given its size and the variety of activities available, Cork is a little better for family visits. Galway's Latin Quarter can be very busy and full of party people, which may overwhelm children and parents during a weekend night out on the town.
If you can only include a one-day visit to Cork or Galway in your itinerary, I suggest that most travelers choose Galway. If you choose Galway or Cork as a base for a longer trip, I suggest that you seriously consider Cork. In Killarney you'll see much more than in either city, and you're close enough to visit Dingle, or even Cork if you wanted to go from there (although I don't recommend Cork, unless you're just going to see Blarney Castle).