The standard for qualifying for unemployment is generally a demonstrated ability and willingness to seek and accept suitable employment. The first hurdle for qualifying for SSD or SSI is the inability to engage in Substantial Gainful Activity. If you are collecting you are representing that you can, and are willing to work. This is contrary to the position that you cannot work due to a disability.
This answer is not meant, nor should it be construed to be legal advice, Further, it is not intended to, nor does it create a lawyer/client relationship with any firm or individual to whom this answer is communicated to.
You can apply for Social Security Disability benefits (both SSI and SSDI) if you are unable to work due to a disability even though you are currently receiving unemployment benefits. Nothing prevents you from putting in the application, even though your statements to each benefit program are seemingly contradictory. However, what may happen is an offset of your retroactive SS benefits based on the unemployment benefits you actually received. This usually happens with SSI retroactive and not SSDI retroactive. As the application process is lengthy, I would consult a SS attorney immediately to get the process started.