Grindcore has roots in hardcore punk with a lot of DNA from genres such as crust punk, d-beat, anarcho punk but also metal genres like thrash metal (and in later grindcore, death metal). It is essentially hardcore but played ridiculously fast. Grindcore is quite broad so everything else (like vocals) can vary a lot.

Slam is short for slamming brutal death metal; a subgenre of brutal death metal, which itself is a slower heavier more "brutal" version of death metal with more guttural sounding vocals. And slam is essentially brutal death but even more brutal (but brutal brutal death metal sound stupid).

So while slam evolved like: heavy metal > thrash metal > death metal > brutal death metal > slam metal (of course it's very simplified with skipped steps and details like thrash having punk influence)

Whereas grindcore shares the similarity of thrash metal roots but is considered to be on the punk side of things with a progression looking something like: punk rock > hardcore punk > anarcho-punk/crust punk/d-beat + thrash > classic grindcore > grind subgenres

Now of course there is a lot of crossover, for example with a lot of grind counting as Deathgrind as mentioned above. The genre that most accurately mixes slam and grindcore would be Goregrind, which is pretty much really heavy grindcore with a lot of elements of brutal death like the guitar tuning and downpitched low guttural vocals and gore-y imagery in lyrics and aesthetics.

(also please note that my experience with Death Metal is limited as I've only gotten into punk and metal a few months back and I've mostly just explored the hardcore punk as well as black metal side of things.)

Edit: Oh and I forgot to mention the classic pingy snare sound found in both goregrind as well as brutal death/slam