Lets be honest, I like them both but Drake and Kendrick Lamar are very similar. Both rap about nothing worth wild, material/subject matter are lack luster, flows are good, both are constantly featured literally on everybody's record, radio equally plays their song, should I go on?
Kendrick calling Drake out was laughable because they are one in the same.
Also everyone keeps talking about Kendrick killing and outshining rappers, when he got killed/outshined by Drake on his own track Poetic Justice.
Lets not act like Kendrick could've gotten this popular on his own. Kendrick needed Drake to push that hit that created buzz for his career now.
For the record I like Kendrick Lamar but I don't think Drake nor Cole should've been called out.
Long response, sorry in advance

Kendrick is one of the most topically versatile rappers in the game, mainstream and otherwise. Dude talks about generational substance abuse (alcohol, prescription drugs, etc., and both from the POV of a user and person watching others' addictions), peer pressure, gang
culture (not just gang violence), domestic violence, prostitution, spirituality, racial profiling by police - and he does all of these things in vivid, candid ways. He really tackles different sides of issues for his generation. Then he can also do the fun/ignorant stuff, standard relationship shyt, etc.
Drake isn't as topically complex, but he's really good at the topics that he does cover, which are his relationships - romantic, friendships, family, and otherwise - and the highs and lows of his fame, and the journey it took to get there. He covers his topics vividly as well.
I do see similarities, in the sense that both good kid, mAAd city and Take Care are essentially coming of age stories as each of them progress from one stage of life into the other.
I personally don't think Kendrick got outshined by Kendrick, but whatever, to each his own. The single helped, but for the most part, his buzz was built by steadily building a fan base with great music and sticking to his guns creatively. Single was definitely a big help for him, as was appearing on "Take Care," but let's not act like the Drake affiliation was the main element in getting him here. Just about every successful rapper worked with another star who helped him get there in one way or another.