Many of the boxed products contain ammonia, which is very hard on hair and can damage it easily. It can make your hair feel like "straw." The salon quality products use peroxide which is much gentler on hair. Another difference is the concentration of the chemicals. A colorist can use the specific formulations that work best for your hair.
You can buy the salon products yourself from a beauty supply store, mix them, and apply them to your hair. It's as easy as the boxed hair colors you are already using, once you know what to purchase. Depending on where you shop, it can be slightly more expensive than the other. Even still, it's much more economical than paying someone.
For example, you could buy Clairol Professional Premium Creme Hair Color in the color/colors of your choice. Then you would buy the Clairol Professional Premium Creme 20 Volume Dedicated Developer. I was told a 20 lift was good for most people.
The boxed hair color was tearing my hair up. It was dry and breaking off. Being a member of a family of nine on one income, I am not going to use money to go to a salon. So, I had to learn how to do it myself. I have been using the salon quality products for four months and already see a big improvement in the condition of my hair.
I am not a hair color professional. I am a woman who does her hair color herself and have learned from others what I shared here. I also have a daughter in cosmetology school who helped me learn what products to use. If your hair is much different than mine, this tip may not work for you. You might need a stronger developer. Do your research, learn, and then experiment...if you dare.