Cold press cooling will reduce blood circulation causing less irritation whereas heat cause will faster blood circulation
When I had toothaches a hot compress always worked best for me because anything cold made me practically scream in agony. Guess it depends if the tooth is abcessed or not, mine usually were. Also, I gargled with warm salt water once every hour and I also put a whole clove alongside the sore tooth.
When I had a toothache, my doctor advised me to apply cold press than the heat one. However, the doctor would need to see your teeth to decide which one worked better for you. Also, he suggested that I use lemon because of its strong acidity that is helpful in the efforts to get rid of toothache fast.
You can click on Home Remedies for Toothache to seek for more useful treatments.