Great question! Most genetic diseases are usually due to a 'loss of function' (or LOF for short) of the gene, caused by a mutation. As an analogy, imagine a mutation is like a spelling mistake in a recipe, the recipe being the gene. So there can be different mutations that cause the same disease, if those mutations cause the gene to not work any more. Depending on the disease, the specific LOF mutation is less important than the fact that the outcome is the non-working gene.

For conditions with autosomal recessive inheritance, an affected child from two carriers with the same mutation (as you mentioned above) would be considered "homozygous" for the mutation i.e. both copies have the same disease-causing mutation. But for most recessive conditions it is common for some affected individuals to have two different mutations but still have the disease: this is called "compound heterozygous".

Hope this answers your question!