How do handling characteristics depend on flying with a full vs. an empty tank?
All aircraft are subject to performance changes due to fuel burn, and since weight, lift, thrust and drag are all tied together for every aircraft in the same way the changes scale nicely. Let's look at some extremes: 

One of the most interesting examples of this was the Concorde. Since the Concorde flew ~20K above every one else in the sky in those days it didn't have the same altitude holding restrictions that regular airliners (or any IFR airplane) has. The Concorde was give a block altitude basically to do what it wanted up around 50,000 feet (ca. 15 km). Since it burned quite a bit of fuel at Mach 2.1 and gradually became lighter during the flight the pilots let the plane slowly drift up to gain efficiency as well as gliding distance if needed. 

The Concorde also used its many fuel tanks to trim the aircraft as well as cool the nose cone. Fuel burn was closely monitored and constantly pumped around to ensure the aircraft remained in trim. 

Even the dinky Piper Archer I fly is affected by partial tanks. If I fly the tanks from full to VFR legal reserves, the CG moves forward a full inch. Granted this does not seem like a lot you will need to trim to compensate for this throughout the flight. You will also be landing a full 240 LBS lighter which is about 10% of the total weight from takeoff. If you come in hot and high you can get yourself into trouble.  

Yes you can feel it, generally pilots are not going to do “faster” maneuvers but the maximum G load may change (see your aircraft's POH for more info). 

Any smart pilot should practice flying their aircraft at various fuel loads to understand the differences it makes in performance and handling. Some aircraft with things like tip tanks or auxiliary tanks even have fuel burn limitations that define where the fuel is to be burned from and in what order, again going from something as small as a Bonanza all the way up to the Concorde as discussed above. 

Just how much can the handling change? Assuming you did your due diligence in pre-flight and calculated your W&amp;B on both ends of the flight accounting for fuel burn to ensure you are within your CG limits the flight should occur with in the limits of the aircraft's POH and generally speaking within operable limits of the airframe. FAA certified aircraft have control force limits so even at the extents of the CG range you should be able to trim the aircraft to handle “normally”. You will likely notice pitch differences (at either extreme) as well as various performance number differences.