Harry Shaw, Dictionary of Problem Words and Phrases (1975) offers this brief discussion of how the two terms differ:
congenial, genial. Congenial means "compatible," "allied in spirit, temper, and feeling," "suited to one another": "The players on this team are congenial." "At the party you will find a congenial atmosphere." Genial means "cordial," "cheerful," "sympathetic": "Our host was in a genial mood. Genial also means "favorable for growth or comfort": "They enjoyed the genial climate of Florida." "A group of genial persons is likely to find that they are congenial with each other."
I must say that I am far more familiar with the "cordial, cheerful, sympathetic" sense of genial than with the "favorable for growth or comfort" sense.
Merriam-Webster's Dictionary of Synonyms (1984) is so far from considering the two words close synonyms that it puts them in separate groups of synonyms. Congenial appears in a bundle with consonant, consistent, compatible, congruous, and sympathetic:
Consonant, consistent, compatible, congruous, congenial, sympathetic are comparable when they mean being in agreement with one another or agreeable one to the other. ... Congenial is most often used of persons or things that are in such harmony with the taste of a person that they afford him pleasure or delight or satisfaction.{a congenial companion} {a pair of not very congenial passengers—Conrad} {the reticence and understatement of of the method made it specially congenial [to the Chinese]—Binyon} {[Hobbes's] theory of government is congenial to that type of person who is conservative from prudence but revolutionary in his dreams—T. S. Eliot} {the ideal of a Greek democracy was vastly congenial to his aristocratic temperament—Parrington} Occasionally congenial is used of things in the sense of wholly and satisfyingly congruous {all such introduced ideas are congenial to the subject—Alexander} {statement, overstatement, and understatement in letters given a congenial context, every one of them is right—Montague}
And genial appears in a bundle with gracious, cordial, affable, and sociable:
gracious, cordial, affable, genial, sociable are used to describe persons or their words or acts who or which are markedly pleasant and easy in social intercourse. ... Genial sometimes emphasizes cheerfulness and even joviality. Often, however, it stresses qualities that make for good cheer among companions (a warm human sympathy and a fine sense of humor) {a genial host} {he was no fanatic and no ascetic. He was genial, social, even convivial—Goldwin Smith}
So on the one hand you have congenial as a characteristic emphasizing compatibility, and on the other hand you have genial as a characteristic emphasizing good cheer. As Shaw suggests the "spirit, temper, and feeling" that congenial people have in common need not be cheerful in the least; it need only be shared. But genial is intrinsically about good spirits, a sociable temper, and affable feelings.