Actually they are not exactly the same.
They can seem to have the same meaning in many contexts, but not in everyone of them.
Since I'm spanish I should be able to explain it, but I'm not sure if I'll do it well enough to be understood:
In the examples you have comment, about food, a "complemento" is something you can add to a food so you can make it more nutrient, better, healthy, etc. It is, you MAY add that to the food, and maybe the result will be great, BUT, there was no need to add it for it to be complete. E.g.: You can add soy to a chocolate milkshake in order to make it more healthy, but there was no need to do it to have it complete as such.
In the other hand, you have a "suplemento". You use this term when you want to mean explicitly that something was missing to make something complete. E.g.: If you are talking about a healthy diet and you say: << A good "suplemento" to make your breakfast a healthy one would be orange juice >>. You are saying, not that the breakfast is "balanced - complete" and would be "more palatable" adding that juice, but that the breakfast is NOT COMPLETE (in balanced terms) and that with that "suplemento" it will be.
A "complemento" for a well-balanced breakfast, would be some aliment you CAN add to it for the reasons you want, and it will keep on being a well-balanced breakfast, but it was already complete.
A "suplemento" for a well-balanced breakfast which still isn't would be something that you NEED TO add to make it well-balanced, or at least "more well-balanced", "more complete".
I'm not sure if I have achieved to explain myself well enough; any doubt, please ask me.