You will have to approach the appropriate authorities, through the school, and get the surnames in both the certificates to reflect the same surname. If you continue with the mistake, in future, you will land in trouble when you join any educational institution.
A Notary Public needs to certify your correct surname. With this document, along with the Aadhar card which should carry the correct surname, if you approach the concerned authorities, this can be corrected. It does take some time and you need to route the request through proper channel, that is, the school where you studied.
Nowadays, everything is computerized, and it should not be a big problem. If the process is delayed, you can still show your original Aadhar card to the Government authorities, if you get a Government job. Whatever be the job, all that the Government wants, is proper documentation. Hence, please do get your certificates corrected, as mentioned above. If you get this done, I do not foresee any problem. You can, in the meanwhile, get the Notary Public and a very well known person in the same location that you are the concerned person, and this should be okay with the Government, till the certificates are corrected.