No. That's the simple answer. There is 3 major differences and 1 typical result.
I know for a fact that I can not apply color to myself with any sort of precision or grace. I will admit... when I was young I tried box color and even Sun-In, just horrible. When I look back at pictures I cringe, my awkward teenage self with blotchy hair. Not cute. The reason why it was so blotchy was because I was trying to apply it. Stylists go to school to learn many different techniques, one of those techniques is how to apply color. Applying color is an art and there are countless ways to create new looks just by the way the color is applied. Box color cannot do that. Even if your friend or family member tried to help you it is still not going to look the same as if a professional applied it.
There is a reason why the drug store color cheaper than salon color, it is because they use cheaper ingredients. You might think you are saving yourself money but when you consider the quick fade and the possible need to pay for a color correction, getting it done right the first time is well worth it. We have had countless "emergency calls" from people who have basically ruined their hair at home and are in tears.
Yes, some of the shiny boxes claim to be "Healthy" , "Ammonia Free", and even "Long lasting"... compared to what? That's the question you should be asking. Cheaper products mean fading faster, damaging, drying, and basically ruining your hair. Every time you sit down in your stylists chair they look at your hair and decide what color to use and many times it's different than the last time you were in. This is because you now have re-growth at your root and that is uncolored hair, where at your mid section and ends of your hair still have color from your last visit. Box color can't do that. The box is not asking you what products you are using, or if you liked your color last time. It's one size fits all, which is not a realty for hair color. The typical result? 9 times out of 10 you will not look like the person on the box.