They are not the same thing. The FMLA (and its California equivalent, the California Family Rights Act), are laws which provide job protection for qualified employees for up to 12 weeks for such things as a serious personal injury or illness, baby bonding time or to take care of a close family member who is seriously ill. Employees who miss work while on FMLA/CRFA are not paid wages, once their accrued sick and vacation pay runs out, unless they qualify for SDI or short term disability.
Short term disability is an insurance policy some employers provide to employees, as a fringe benefit, whereby the insurer will pay a portion of the employee's wage if the employee is unable to work due to an illness or injury which occurs during the coverage period. It is not a legally mandated program and there is no job protection because short term disability plans are not designed for that purpose. It does not apply when an employees needs to take time off to care for a sick family member.
They say you get what you pay for, and this response is free, so take it for what it is worth. This is my opinion based on very limited information. My opinion should not be taken as legal advice. For true advice, we would require a confidential consultation where I would ask you questions and get your complete story. This is a public forum, so remember, nothing here is confidential. Nor am I your attorney. I do not know who you are and you have not hired me to provide any legal service. To do so would require us to meet and sign written retainer agreement. My responses are intended for general information only.