The main reason for Republicans claiming to prefer Putin to Obama is partisan politics. People pledge allegiance to one party and defend that party at all costs. In this case, the Republican Donald Trump spent a lot of time blaming Obama for America's problems and a lot of time praising or defending Putin, so a lot of Republicans began to think that maybe Putin isn't so bad. Another reason is racism. A certain portion of Republicans hate President Obama because he is the first black president. He obviously conforms more closely with most of their beliefs than a totalitarian such as Putin, but because he is a Democrat and an African-American, they automatically hate him.
Republicans have found themselves swooning over Donald Trump, hanging on his every word and idolizing everything he does. When Trump suggests he admires Putin, the Republicans find themselves admiring him too. It really has nothing to do with what kind of person Putin or Obama is, it is solely because of the near brainwashing that Trump has done.
I'd put down Putin's popularity over President Obama to a lot of lingering resentment, fake news and racism. Or maybe a lot of ignorance. For some stupid reason or reasons people in this country are failing to realize that Russia isn't our friend and Putin is a dictator who treats his enemies, or even just dissenters in pretty brutal ways. No way should this person be more popular than our current President.
Certainly the Kremlin didn’t like Hillary Clinton. Putin has fashioned himself as a defender of traditional values around the world, something that has a particular appeal to the socially conservative elements of the Republican Party. He’s actively pushed anti-LGBTQ and anti-abortion legislation in his country. Just this week, the Russian government prevented the UN Security Council, in their statement about outgoing UN Secretary-General Ban Ki Moon, from praising the Korean’s promotion of LGBTQ rights during his time in office.