Good question! They are not synonyms. I will try to explain the differences the best I can.
The first difference is that "Allegation" is a noun, and "Blame" is usually a verb. Blame can also a noun, but I think it is more common to use it as a verb. "Allegation" means "A claim that someone is guilty" (noun) and "Blame" means "to claim that someone is guilty" (verb). So the differences between your two definitions come down to the difference between claim as a verb and claim as a noun. Google gives the verb definition as
State or assert that something is the case, typically without providing evidence or proof.
and also gives the noun definition as:
An assertion of the truth of something, typically one that is disputed or in doubt.
So if I were to Claim (verb) that I just won the lottery, I stated or asserted that I won the lottery. I also made a claim (noun) that I just won the lottery. I made an assertion of the fact that I won the lottery.
Now as for "blame" as a noun, the differences are more obvious. "Blame" as a noun is synonymous with "guilt". If someone makes allegations that I am guilty, it is still a possibility that I am innocent. That only means that someone says that I am guilty. The same is true if someone blames (verb) me. However, if "The blame is on me" or "I take the blame" (both nouns) then it does not leave an option that I am innocent. It means that I am guilty.