THEY ARE LISTED AS SYNONYMS WHICH MEANS THAT THEY MEAN THE SAME THING USUALLY. HERE ARE SOME DEFINITIONS FOR BOTH.
as·sume (-sm)
tr.v. as·sumed, as·sum·ing, as·sumes
1. To take upon oneself: assume responsibility; assume another's debts.
2. To undertake the duties of (an office): assumed the presidency.
3. To take on; adopt: "The god assumes a human form" (John Ruskin).
4. To put on; don: The queen assumed a velvet robe.
5. To affect the appearance or possession of; feign.
6. To take for granted; suppose: assumed that prices would rise. See Synonyms at presume.
7. To take over without justification; seize: assume control.
8. To take up or receive into heaven.
pre·sume (pr-zm)
v. pre·sumed, pre·sum·ing, pre·sumes
v.tr.
1. To take for granted as being true in the absence of proof to the contrary: We presumed she was innocent.
2. To constitute reasonable evidence for assuming; appear to prove: A signed hotel bill presumes occupancy of a room.
3. To venture without authority or permission; dare: He presumed to invite himself to dinner.
v.intr.
1. To act overconfidently; take liberties.
2. To take unwarranted advantage of something; go beyond the proper limits: Don't presume on their hospitality.
3. To take for granted that something is true or factual; suppose: That's the new assistant, I presume.