As people above have mentioned, I think the main problem lies with standardization. The ICD helps provide a coding standard in documenting diseases, such that different healthcare providers or peripheral stakeholders such as researchers and insurers could have a consensus of coding in referring to the same disease. If the ICD is not used, there could be confusion and misunderstandings between different entities.
The ICD also helps track the incidence of diseases by providing a clear categorization of diseases such that we could easily count the occurence of certain conditions (for example, all codes starting with F are mental conditions). Without the code, it would be really difficult to compile the statistics since we would only document the condition by name and the data would be very unstructured (depression, schizophrenia, etc.). With such a code, the data becomes much more structured and we can sort things since codes of the same initial letter usually belong to the same nature. This helps public health surveilance and epidemiological research tremendously.