For the tires, concrete is a non-yielding non-compressible surface so over time the tires will become flat in those areas. If you have a finished floor you would see the tire tread become part of the floor surface when you do move the car. Also at that spot on the tire it will be no longer round. You d hear this when you drove the car and everytime you go to that flat spot you d hear "b-lap, b-lap, b-lap". It may go away but you would probably have to have the tires re-balanced and redistribute the wheel weights to compensate for your flat spot AND where the tire was glued together at the factory. The wood is a compressible medium under the weight of the car allowing the tire to retain its shape(also keeping tire treads from appearing on your finished floor). The surface of the wood takes the pressure from the tire.
Can you cite to some reputable source that supports the premise conerning putting wood under tires "allowing the tire to retain its shape."
For the battery, the wood keeps the temperature of the floor from affecting the battery case. All car batteries now are Maintenance Free. They dont require distilled water to be added but they still have the same vents on them under the "sealed" caps. If you put the battery on a concrete floor and the floor gets hot from immediate sunlight, because concrete is essentially like stone, then the heat could transfer to the bottom of the case and then you could have electrolytic loss or cell sulfation. The wood acts as a minute temperature buffer between the floor and the battery AND also keeps your floor from suffering the effects of sulfuric acid if the battery vents excessively while you are away. And dont forget the case is plastic and it can crack because hot weather battery have thinner cases than cold weather batteries. Do a simple search on the Sears Auto website and you will see that they offer every battery in a North environment and a South environment. The difference is the case.
Your premise on concrete heating up would be correct if the concrete was exposed to direct sunlight. However, I believe the garage of the OP is enclosed and in such case the coolest portion of the garage would be at or on floor level. In her case, the concrete floor would actually be cool to the touch and ideal for preserving battery life. So putting the battery on wood may raise the temperature....exactly what you do not want in a tropical climate.
And while I agree that a cracked battery case, leaking battery acid, on a concrete floor would cause the battery to discharge faster, that is a rare exception.
And while I agree that northern batteries have thicker cases than southern batteries, (note the OP shipped her car from near home NY/NJ so presumably it has a northern battery, the difference is not just the case.
Northern batteries contain more internal plates than southern batteries. The reason and the real difference in the two types of batteries is CCA's. Batteries made for northern climates need more cold cranking amps to start the car in severe weather. That requires more internal plates....and thus a thicker case to hold the extra plates/extra weight.
The difference is way more than the case.
Respectfully,
Playacaribe2