It generally won't be more helpful. Not only will the names be different in different countries, there may be different types of the same species with different properties, or even different species with the same common name. If you have an allergy to something like coriander, being able to read the label and see what contains actual coriander and what contains curry leaf is essential.
Lavender is a great example. Any gardener could tell you that English lavender (angustifolia) is a very different plant from French lavender (dentata), even though they're the same family. I've no idea what the essential oils will smell like, but I'm pretty sure they won't be identical. And more importantly, if you're talking allergies then I'm pretty sure they won't have the same chemical composition.
Even worse, consider bergamot. Bergamot oil should come from the skin of the bergamot orange, but there's also a species of flower (monarda) which is commonly called bergamot because its scent is similar to bergamot orange. If your contents just said "bergamot", which one? And even worse, there are two types of monarda which are completely different plants.