I think that in some ways there are clashes in the jurisdiction of the two court systems, but in other ways they both keep each other honest.  The overlap in jurisdiction allows the two systems to work together when they can.  It is true that there are times when political or other considerations lead to disputes between local and federal investigations and trials, there have also been cases where the cooperation of locals and feds led to quicker or more appropriate justice.

I think that it is important for each state to be able to address issues that are unique to them take the issue of immigration like it or not the state of Arizona has determined that this is a problem they want to deal with in a specific manner. I think states have to be given that option.

I can see two ways the justice system would be different if the state and federal courts were in a unified system.

1) I think the laws now enforced by the federal government would be more unequally and ineffectively applied.  It would be more difficult to have a national law effectively enforced, in other words.

2)  I also think what are now federal crims: kidnapping, treason, etc. would get thrown in with the massive backlog of state crimes, and justice overall would be less swift.

The answer by pohnpei397 is a better answer than mine, but I think I do have some contribution to make in the way of food for thought.

If there were no federal court system, the laws and legal systems might better meet the needs of the various regions of the U.S.  Rather than a one size fits all, there might be a good fit for each.  If each state did its own thing, then each state could compare its achievements with the achievements of 49 other states and adopt from each what was best.  With one single court system dominant over so much diversity as exists in the United States, it would be a great accident if the system were a great one rather than a mediocre one.  It has nothing to compare itself to except imagination.

This is, of course, a matter of conjecture because we cannot know what would have been.  However, I think you could argue that our criminal justice system would be more of a patchwork than it is now.  There would be states with much more or much less in the way of protections for defendants than is currently the case.

I think that more aspects of our system would resemble the death penalty -- with wide divergence among the states.  We currently have states with and without the death penalty.  We have states with the death penalty that execute relatively few people and states that execute many.  This is partly because the federal government has stayed out of it and let the states have a great deal of leeway.

So I think that we would have more of that.  We would have states where the exclusionary rule would be weaker or even nonexistent.  We would have states where there would be more (or fewer) exceptions to the rule requiring a warrant for searches.

Without the federal court system to oversee the states, I think that there would be more differences among the states with more conservative states being much "tougher" than more liberal states.