Turf is a quick fix but takes a lot of looking after for the first few months. The roots need to regrow so it needs to be laid properly with no gaps, and constant watering in if needed.
Grass seed will take longer to give a good thick covering, but is much cheaper and easier. Once the seed has sprouted, it doesn't need much care but won't tolerate heavy wear from feet or hungry buns for a while. You also know that it hasn't been chemically treated if the buns are going out on it.
I'm in the same boat - the lovely fine lawn has taken a hammering over winter and I now have a large muddy patch with moss and dandelions. I'm going to loosen it all up with a fork (stick it in and gently rock to aerate every couple of inches, rather than dig it over), smother it in sand, brush / rake level and reseed. The buns will have to be kept off each area for a few weeks, but it will take me that long to do the whole thing properly.
The grass has started to grow now so it is a good time to either turf or seed, before the summer weather and hosepipe bans set in.