Historical Differences:
Grindcore developed directly from crust and hardcore, death metal borrowed some techniques from hardcore (Discharge) but is largely metal-derived.
Conceptual Differences:
Grindcore retains the politicized nature of hardcore, as well as many of its explicitly deconstructive leanings. While death metal has occasionally touched on political and social issues, it tends to be more indirect in metal's Romantic fashion, focusing on existential chaos, occultism and gothic horror.
You do see some lyrical overlap, as both genres frequently touch on death and gore as grist for the thematic mill. However, they handle these topics in a rather different fashion, with grindcore "playing" (in the postmodern sense) with death and pain as value signs for didactic purposes (Carcass) and death metal approaching them as an exercise in horror (Autopsy).
Structural and Aesthetic Differences:
Grindcore generally retains the pared down, strictly linear structures of hardcore and crust, as well as the rhythmic and melodic simplicity (both within phrases and songs as a whole). Death metal is heir to the "riff salad" approach common to metal (and derived ultimately from prog rock), and tends to not only be more complex rhytmically and tonally, but more prone to recursions and structural meandering. As a result, grindcore songs tend to be very short (under 2 minutes, often much less) and devoid of the radical tempo and melodic shifts found in most top tier death metal. Grind is also tends to retain the DIY production aesthetic and ironically sloppy playing of hardcore, in contrast to the clinical precision, professionalism and Romantic dedication to craft and atmosphere one typically finds in death metal.
Obviously, at this point there has been so much cross-pollination between grind and death metal that a lot of times it comes down to social standing and who bands associate themselves with.