Palliative care focuses on symptom relief and quality of life, and can run alongside curative treatment at any stage of an illness. Hospice is for a life-limiting illness when a physician expects six months or less if the disease runs its normal course, and generally means curative treatment has stopped.
The distinction that matters
Families frequently treat 'palliative' as a synonym for end-of-life care, and that misunderstanding sometimes delays a referral that could have helped much earlier -- palliative care is available at any point in a serious illness, not only near the end. Hospice, by contrast, requires a physician's certification tied to a specific prognosis, and choosing it does not have to be permanent: a patient can revoke hospice and return to curative treatment, or switch hospice providers, if circumstances change.