The material the dentists use to fill up the cavities in teeth should have the same coefficient of expansion as the tooth itself what happens if they are different?

All dental filling materials have a different coefficient of thermal expansion than natural tooth structure. Some are closer than others, but none are identical. A couple of things come into play -- one is that teeth flex slightly -- if teeth were totally brittle, and had no "give" they would always break as soon as there was a temperature difference. A second factor is that coefficient of thermal expansion depends on the amount of material -- i.e. larger/wider/longer fillings will shrink and expand more than smaller ones. In practice, fillings tend to be fairly small, so the expansion/contraction is small, and usually not clinically significant. The third major factor is that they still do expand/contract at different rates, and sometimes the result is microleakage. Microleakage is the leakage that happens between the filling material and the liner or adhesive layer that is on the tooth. Again, this is typically not a clinically significant problem, and has been researched quite extensively. The closest class of materials in terms of coefficient of thermal expansion are glass ionomer materials, followed by gold, then amalgam, and the worst is composite (at least in terms of thermal expansion). When choosing a material for a restoration, there are multiple factors, and there is not a "one size fits all" or a single best material that is perfect in all cases.