It depends on what you consider "nothing" to be. If you think of it as a specific state of quanity of something, then zero is certainly "nothing". If my bank account has a balance of zero, then I could say "I have nothing in the bank".

However, if you treat zero as an element or object itself, then "zero" and "nothing" can take on different meanings in mathematics. If I asked "What are all the numbers which when multiplied by 5, equal themselves?", the answer is "zero" because that's the only number. If I asked "What real number times itself is equal to -5", the answer is "nothing". The solution set of the first is {0}, but the solution set of the second is {}, also known as the "empty set".