In general, Jefferson Davis had the same powers as the United States President. The Confederate Constitution, barring some changes to protect slavery and weaken the central government, was largely a copy of the US Constitution. There were some changes, but the most important one was that the President could only serve a single six-year term, elected by an Electoral College that worked like its US version.

When it comes to the President's powers, he was declared to be the Commander in-chief, and was given the power to appoint executive officials. The Confederate Constitution adds a clause that explicitely gives him the power to remove any executive official whenever he wants. However, the US President could do the same; it wouldn't be until the Tenure of Office Act that limits on this power were tested, and then declared unconstitutional.

Thus, Davis largely had the same powers as Lincoln, and used them in largely the same way. Whether Davis was an objectively commander in-chief is difficult to ascertain, because Davis had to work with different resources and people. Nonetheless, he in several occasions used war powers just like Lincoln did. He, for example, suspended the right of habeas corpus with the consent of Congress in order to enforce conscription, and interpreted the "necessary and proper" clause of the Constitution to permit conscription. This seems paradoxical, since the Confederates were supposedly fighting for states rights and a weak Federal government (aside from, you know, slavery). But as McPherson says, he had to "don the mantle of Hamilton" to achieve the Jeffersonian objectives of the Confederacy.

In general, Davis' war powers were more limited than Lincoln. Part of it was the aforementioned ideological reasons. But Davis also had to face a much more hostile opposition, which often was based on personal grounds. It cannot be denied that Lincoln was a more skilled politician. Though Davis was not the arrogant monster that some adversaries and historians portray him as, he could not forget grudges or stroke the egos of opponents, thus gaining their unwavering enemity.

He also did not have a political party that could allow him to marshal support for the administration like Lincoln did. While Lincoln could count with the support of the Republican party and their majorities in Congress and the state governments, Davis was actively opposed by the anti-administration men in the military, the states, and Congress. For example, the Confederate conscription law included a clause exempting certain jobs and state officials from the draft. Anti-administration governors quickly appointed thousands of new officials to keep them from being drafted. This has led some to declare that the Confederacy "died of states rights." (Edit: Lincoln also had to deal with a nasty opposition, but Davis' allies were much less united than Lincoln's, and for Davis' enemies it was often a personal quarrel, while when it comes to Lincoln it was almost always strictly ideological or factional.).

Likewise, Davis used the suspension of habeas corpus much more sparingly than Lincoln did. And while Lincoln suspended the writ on his own authority, Davis only did so when Congress authoritzed him. Nonetheless, Mark Neely concludes that “Abraham Lincoln and Jefferson Davis acted alike as commanders in chief when it came to the rights of the civilian populace,” and that they "showed little sincere interest in constitutional restrictions on government authority in wartime."

Compared with Lincoln, Davis was almost entirely focused on the military aspect of the war, trusting his Secretaries and Congress to oversee its economic, diplomatic, and social aspects. He thus had a much more "hands on" approach to the war than Lincoln did. But at the same time Lincoln also took charge of social (such as Reconstruction or Homestead bills), economic (measures creating the greenback) and diplomatic aspects (foreign relations with Britain) that Davis did not concern himself with, even though he undoubtedly had the right, power, and duty to direct.

To summarize, Jefferson Davis had constitutionally the same powers as the President of the United States did, but greater opposition, his ideological convictions, and the realities of the Southern war effort means that he did not interpret and use those powers as broadly as he could, unlike Lincoln who often used the justification of "war powers" to massively expand the national executive and the government as a whole.

(Note: I'm not saying that Lincoln was a tyrant with disregard to the Constitution. But he did interpret his own powers more broadly than Davis did, and was much more involved in other aspects of government outside of the war).

Sources:

Mark E. Neely. The Fate of Liberty.

James M. McPherson. The Battle Cry of Freedom and Embattled Rebel.

And, of course, the US Constitution and the Confederate Constitution.