The main difference in policy seems to stem from the US embargo on Cuba. Some info on the embargo from Wikipedia:
The United States currently imposes a commercial, economic, and financial embargo against Cuba. The United States first imposed an embargo on the sale of arms to Cuba on March 14, 1958, during the Fulgencio Batista regime. Again on October 19, 1960 (almost two years after the Cuban Revolution had led to the deposition of the Batista regime) the U.S. placed an embargo on exports to Cuba except for food and medicine after Cuba nationalized American-owned Cuban oil refineries without compensation and as a response to Cuba's role in the Cuban missile crisis. On February 7, 1962 the embargo was extended to include almost all exports.
As of 2018, the Cuban embargo is enforced mainly through six statutes: the Trading with the Enemy Act of 1917, the Foreign Assistance Act of 1961, the Cuban Assets Control Regulations of 1963, the Cuban Democracy Act of 1992, the Helms–Burton Act of 1996, and the Trade Sanctions Reform and Export Enhancement Act of 2000. The stated purpose of the Cuban Democracy Act of 1992 is to maintain sanctions on Cuba as long as the Cuban government refuses to move toward "democratization and greater respect for human rights".
Obama campaigned on and worked toward lifting the trade embargo, from the New York Times, dated October 2016:
WASHINGTON — President Obama on Friday moved to cement his administration’s historic opening with Cuba by issuing a sweeping directive that will last beyond his presidency, setting forth a new United States policy to lift the Cold War trade embargo and end a half-century of clandestine plotting against Cuba’s government.
Trump has opposed this long before Obama became president. In 2016, the Miami Herald reposted an editorial Trump wrote in that paper back in 1999. In it, Trump argues against lifting the embargo.
I'm not going to quote the entire column, but the last paragraph speaks volumes, remember that Trump is the author of this editorial in the Miami Herald of June 25, 1999:
Yes, the embargo is costly. If I formed a joint venture with European partners, I would make millions of dollars. But I'd rather lose those millions than lose my self-respect. I would rather take a financial hit than become a financial backer of one of the world's most-brutal dictators, a man who was once willing to aid in the destruction of my country. To me the embargo question is no question at all. Of course, we should keep the embargo in place. We should keep it until Castro is gone.
At the time, Trump also sat down with Chris Matthews of MSNBC. I will quote part of the transcript as listed on factba.se:
Chris Matthews: Let's talk about the commander-in-chief's job, which is to many people the most important job of the presidency when we're in a war. In our lifetime, we've had a lot of wars. I'm going to ask you -- I know this is -- oh, the people at National Public Radio will say this is unsophisticated, but I want to ask you the simple question: Did you support these military campaigns when you were thinking about them at the time and in retrospect? Or don't you? Was the Bay of Pigs a good idea?
Donald Trump: I think it was, but I think Kennedy let everybody down very seriously. I think that you wouldn't have had a Fidel Castro had Kennedy done the bombing.
Chris Matthews: Gone all the way with air cover?
Donald Trump: Well, I would have done it. I mean, at the time, I would have done it. In retrospect, obviously...
Chris Matthews: With the flag flying?
Donald Trump: He should even gone further than that.
Chris Matthews: You would have gone with the flag flying, supporting the Cubans?
Donald Trump: I would have supported the Cubans. They would have had a free Cuba right now, and we wouldn't have gone through many, many years of Fidel Castro.
The embargo still isn't of the table for Trump. Only a few months ago, President Trump tweeted:
If Cuban Troops and Militia do not immediately CEASE military and other operations for the purpose of causing death and destruction to the Constitution of Venezuela, a full and complete....
....embargo, together with highest-level sanctions, will be placed on the island of Cuba. Hopefully, all Cuban soldiers will promptly and peacefully return to their island!