I buy a lot of CAFO beef due to limited funds; I use grassfed tallow with that. My primary use of coconut oil in cooking is for poultry; I melt the oil and brush it on the poultry skin. I used some last week for baking a whole fish, too, although I also wedged a chunk of butter into the abdomen.
When cooking vegetables, I'll use either tallow or butter because I find I'm very good at tasting the coconut oil and it's distracting. My friends say they can't taste coconut oil but I sure do. I've tried melting a little into coffee but I found the flavor incredibly "wrong" so now I stick with heavy cream.
If you're using high heat, coconut oil handles that the best; with low heat, it's hard to beat butter.
I use beef fat and butter. :)
Coconut oil has no cholesterol and is very high in salicylates. For those with sensitivity to salicylates, coconut oil is not an option for food choices.
Here is an article at WAPF on cholesterol, by Natasha Campbell-McBride M. D.
And here is Chris Masterjohn's site, Cholesterol and Health:
The super-high melting point of tallow is something of a turn off for me. I often have it solidifying on my food as it cools down which is somewhat off putting. And as far as taste, tallow is kinda meh to me (especially compared to lard or butter). Although I will admit that onions will never taste better than when sauted in grassfed beef tallow.
I don't like using coconut oil in many foods. The flavor is odd and hard to pair with many foods. I'll use it when baking coconut flour recipes as a butter substitute. Mostly, I just spoon it out and eat it plain though.
I use tallow, ghee or lard (although I'm rethinking the lard), mainly because I'm too cheap to use expensive coconut oil for cooking. One way I do sort of use coconut oil for cooking is to add it to things like paleo crépes* that I make a lot for my son, the oil oozes out and creates a non-stick effect as the crépes cook, plus it's a healthy fat, of course. I agree with Nance that coconut oil has a distinct flavor that can overpower or clash with foods, although it is flavor enhancer for things like crépes.
*I actually make normal crépes for my kid, with actual white flour (apostate!), cause I think the egg, milk/creme, coconut oil and touch of honey make up for the flour and I don't have easy access to alternatives like coconut flour--I put it in the 70/30 category.