There are at least two important similarities between these two presidents.
First, both of these men were born in very humble circumstances and were self-made men. Jackson was born in the backwoods of South Carolina and his father died soon after Jackson was born. Lincoln was not from a poor family, but his father was a farmer and carpenter, not a member of any elite.
Second, both men were strongly in favor of keeping the Union together. Jackson faced threats of secession from South Carolina in the crisis over the "Tariff of Abominations." He acted strongly to prevent secession. Lincoln, of course, was willing to go to war to prevent the South from seceding.
Lincoln and Jackson are similar in that they both strengthened the power of the American President. Lincoln guided the Union to a difficult victory in the Civil War and, in the process, expanded the power of the executive branch. For example, he instituted the first income tax to pay for the war, and he suspended habeas corpus rights in border states to allow suspects to be tried in military courts. In addition, he instituted a draft and issued the Emancipation Proclamation in 1863, which freed slaves in the Confederate states. Jackson expanded the power of the President in part through the spoils system, by which he rewarded loyal followers with patronage. In addition, Jackson took an active role in issues such as Indian Removal and the dismantling of the Second U.S. Bank. He also passed the Force Bill to compel South Carolina to pay the tariff (see the paragraph below for more information).
Both Presidents were also defenders of the Union. Lincoln fought the Civil War to preserve the Union, and he did not believe the Confederate states had the right to secede, or leave the nation. Jackson defended the Union when South Carolina threatened to secede during the Nullification Crisis of 1832. The state refused to pay the Tariff of 1828 (and the Tariff of 1832), referring to it as the Tariff of Abominations. When South Carolina nullified the federal law and threatened to secede, Jackson supported Congress in passing the Force Bill, which stated that the President could use military force to compel South Carolina to follow the law and pay the tariff.
Lincoln and Jackson were also the fathers of the modern political parties still in existence today. Lincoln was the first Republican president, and Jackson was the first Democratic president. Though the parties have changed throughout the years, Jackson is considered the founder of the Democratic Party, and Lincoln was, while not the founder of the Republican Party, the person who first represented Republicans in the executive office.