Large core sizes increase power and depth.  Use large cores when you want maximum power and tone on large timpani.  Higher head tensions generally need a larger core and/or more felt than lower head tensions.

Some types of timpani produce a less focused sound than others and may need smaller cores and/or less felt than other types to give good projection.

Natural skin timpani heads generally need a smaller core and slightly more felt than plastic heads.

Very slack heads and low notes are generally better with cork or hard-felt cores.

Clarity of sound and the acoustic qualities of your concert venue
The exact degree of clarity that your audience hears will be affected by many factors including your timpani and heads, but most significantly by the sticks you use and the acoustics of your concert venue.     

Very resonant acoustics such as cathedrals can make the timpani very indistinct with a very heavy bass quality to the sound.  This type of acoustic needs a more sharply focused, less full sound than usual.  You can achieve this by using smaller cores and thinner felt than usual.  The smaller cores will produce less fullness and depth from your timpani and the thinner felt will give a clearer and more articulate sound.  

Concert halls with very dry acoustics need the opposite treatment and you may wish to select larger cores to maximise the depth of tone and resonance produced by the timpani and use thicker felt to reduce the articulation heard by the audience.

When playing in smaller halls or with a small group of players, you may wish to reduce the fullness of sound by choosing smaller than normal cores so that the timpani are not overpowering.

The perceived hardness of the sticks
The size of the core can affect the perceived hardness of the sound.   For example a 310 stick (small core) will sound harder and more brittle than the larger 610, even though they both use the same 1mm depth of felt.  This is because the 600 core produces more tone and depth than the 300 core and this bigger, fuller more rounded tone is perceived as less hard than the sound of the 300 core.