The main difference between the Lincoln's and Johnson's (presidential) Reconstruction plan and the Radical Republican plan was that the Radical plan wanted to strongly punish the South while the Presidential only wanted to get together the union as fast as they could.
Apart from that, in the presidential plan, they wanted people from the South to take an oath of loyalty, this was called the 10% Plan. It meant that once at least 10% of the state had taken the oath, they could return to the Union as a state.
However, in the Radical Republican Plan, instead of wanting the 10% as the presidential Plan, they wanted 50%.