It is a very large question and it will be difficult (even with all information) to give a strict answer.
Both LEAP-1A (A320neo family) and CFM56-5B (A320ceo family) are turbofan engines which means they have two cores (HP and LP core) and two flows, and where total airflow is created by a Fan. As you said, LEAP-1A has a larger diameter allowing higher by-pass ratio (efficiency improvement) and lower rotational Fan speed (noise reduction). Besides, OPR (Overall Pressure Ratio) is greater (~40:1 vs. ~33:1) and lead to a performance improvement too.
Nevertheless, even if LEAP-1A has a slightly greater max thrust (~155kN vs. ~140kN), we cannot affirm there is "a range improvement" since they do not operates on same aircraft. If it's true here, it is more appropriate to compare aircraft range than engine range. Moreover, thrust is calculated for an aircraft flight mission (by Airbus, Boeing, etc.) and then generally given to engine manufacturers as input for conception.
Now, to answer more specifically your questions, LEAP-1A service entry was in 2016, more than 20 years after CFM56-5B first flight. Therefore LEAP engines gather a lot of improvements resulting from years of research and maturity tests. So yes, there is others differences between these engines (you can find public characteristics on manufacturers or specialised websites - even wikipedia get a lot of information actually).
1. Are the engines lighter than its predecessor (are they made of lighter alloys?)
Material is probably one of the field where you can find best improvements in aeronautics the last decades, for both aircraft and engines. In particular, many advances have been made with composite materials.
When comparing CFM56-5B and LEAP-1A, the most significant (and literally the most viewable) point is about Fan material. For a CFM56-5, fans are probably made of steel or nickel alloy (I don't know exactly but these are common for old generation engines). In a LEAP engine, composites are used for fans with titanium at leading edge. This allow, amongst others, a lower number of blade (18 blades) and larger chord: fan mass reduction despite larger diameter, aerodynamic efficiency, etc.
But is LEAP-1A really lighter than its predecessor CFM56 ? Not sure, remember that the diameter is larger. A quick search on google give 2900kg for LEAP-1A and 2400kg for CFM56-5B, but I don't know if it was calculating in the same way.
Note that other improvements are made in high temperature parts (like HP turbine) both in cooling and material resistance resulting in a higher combustion temperature (again there is an efficiency improvement).
2. Do the new engines have the reduction gearbox which lacks in the ceo engines?
No, both LEAP-1A and CFM56 do not include gearbox. CFM international has never developed an engine with it. This is a manufacturer choice as the gearbox lead to an extra mass and often extra maintenance costs. In the other hands, gearbox give advantages by decoupling LP turbine / booster and fan speed and remove one constraint to optimise each component.
As @Bianfable said in comments, it is a little bit awkward to talk about ceo and neo engines as Airbus manufacture these aircrafts with 2 different engines. For example, Pratt & Whitney PW1000G turbofan on A320neo include a reduction gearbox..
3. And what is TAPS II combustor technology?
TAPS (Twin Anular Premixing Swirler) is a combustor technology developed by GE aviation. While CFM56-5B used SAC (Single Annular Combustor) or DAC (Double Annular Combustor), TAPS is a newer type of combustor used for GEnX engine (Boeing 747-8 and 787). The version II is the scaling version for LEAP engine. The main idea is to limit rich combustion zone (mainly responsible for NOx).
Take a look to this FAA report for interesting details.