Electrolytic capacitors give relatively high values in a small space because the dielectric layers are very thin, and the capacitor contains a very large surface area. The dielectrics are metal oxides, created on the surface of the metal by putting the metal as the anode in an electrolytic cell. The oxides formed are adherent and compatible with the metal underneath, so that they do not ‘flake off’, and the thin layers have a very high voltage withstand, of the order of tens of MV.m–1.
For both sorts of capacitors, the anode connection is direct to the anode foil onto which oxide has been grown. The cathode connection has to make contact over a wide area of oxide surface, and this is done by filling the space between oxide surface and the counter-electrode by a conducting material. Depending on the construction, this may be liquid, gel or a solid conducting or semiconducting material.
The aluminium electrolytic capacitor is made with pure aluminium foil. The surface area is increased dramatically by suitable etching, and the foil is packed into a small space by rolling it tightly; the tantalum capacitor creates a large surface area by being made of a sponge of sintered tantalum particles, carefully selected to have maximum surface area.
Aluminium electrolytic capacitors have liquid or gel electrolytes; those for tantalum capacitors are solid, and either manganese dioxide or (in a more recent development) a conducting polymer.
Tantalum electrolytic capacitors are more costly, but offer superior ESR and greater capacitance density. They are also able to operate over a greater temperature range. However, they are substantially more expensive than aluminium electrolytic capacitors. For power applications, where heat has to be dissipated within the capacitor structure, the aluminium electrolytic capacitor would normally be chosen.
You might also explain to them the differences in the packages available.