Aluminum capacitors are made by separating an inner and outer aluminum electrode by a layer of electrolyte paste contained in a porous spacer. When a voltage is presented across the plates, a chemical reaction occurs at the surface of the aluminum, forming an Aluminum Oxide layer on the surface of the aluminum plate. The Aluminum Oxide is a dielectric, and the layer is relatively thin. Two conductors separated by a dielectric make a capacitor.
When the paste dries out, the capacitor is no more. If you reverse the voltage, the dielectric layer breaks down rapidly, gases can form, and the capacitor will go "pop" (or sometimes "boom").
Tantalum capacitors are made by forming powdered tantalum around tantalum wire. The sintered pellet is dunked in an acid bath and DC is applied across the pellet. Current flow causes Ta2O5 oxide to form on the surface. A maganese oxide layer is formed over the pellet by dipping it into Mn(NO3)2 solution and then heating. A wire is then bonded to the outside of the pellet. A tantalum capacitor has extremely thin dielectric, resulting in easily obtained high capacitance values.
Unlike the aluminum capacitor, there is no electrolyte to dry out. Since it is dry, its lifetime is much longer, and it also has a lower leakage current. Tantalum capacitors can also handle some reverse voltage without failing - but not much. Because of their smaller physical size, the voltage rating of tantalum capacitors is lower than aluminum capacitors.