The main difference is the developer and the application. A box dye cannot see your hair, a colourist can. Most box dyes contain a 6% developer, which is the middle of the road kind. There's really no need to use 6% in most cases, although grey, resistant hair is one of the reasons you would use a 6%, for a lot of grey, an even stronger developer might be used.
Application wise, a trained colourist can correct uneven tones in the hair by using one or more colours and also work with the base shade to ensure the right result is achieved, whereas on a box dye, if you're not the right base shade for the colour, it's not going to turn out how you want it.
There's also things we can do to work with ultra resistant grey hair that would be hard to copy at home.
There's no other real difference in the formulas (in most cases, there are some brands to avoid). If you're happy with the results of box dye, crack on with it. Don't try to go lighter at home. Just don't. Stick with the same level colour and things will work out fine 99% of the time.
The only thing I will say is that while Henna is actually quite good for your hair, it limits your options as to what you can do with it afterwards. It contains metallic salts which coat the hair and react badly with most other chemical processes. If you're not sure you want to keep it until it grows out, don't do it. You can't lighten over henna as it reacts with bleach, you can't use perming or straightening lotions over it and I believe some colours react with it too. Most hairdressers won't touch you if you have henna on. Lush Henna blocks, I've been told, don't contain the same ingredients as other henna and can be lightened and coloured over.
Live XXL is also a bit iffy for leaving shit behind on the hair. It's impossible to remove and coats the hair with metallic salts and silicone (that's how the colour goes so vibrant and stays so long, it's welded to the hair shaft with silicone). If you decided you wanted to change it, you'd be hard pressed to find a stylist willing to remove it for you (colour B4 works, ime).
Sun-in and Just For Men are also brands to avoid as they react with other chemical processes you might want in the future. Other than that, box dyes are perfectly fine and no more damaging than salon brands.