In Russian and other Slavonic languages they are well-established as meaning ‘palatalized’ and ‘non-palatalized’ respectively (which in Irish would be ‘slender’ and ‘broad’). But an English ‘soft’ C is one pronounced [s]. A Hiberno-English ‘soft’ t is a fricative.
Now consider this quote from the Wikipedia article entitled Tikka.
Tikka or Teeka or Teekka is the English transliteration for two entirely distinct Indian words: tikka with a soft initial 't' and tikka with a hard initial 't'. This often causes some confusion as to which "tikka" is meant.
Tikka with a soft initial 't' means a piece of meat, such as a cutlet.
Tikka pronounced with a hard 't' can mean a forehead mark or a needle.
So what is the phonetic difference?
After some research I was able to track down the two Hindi tikkas. The “soft” one, the one we know from chicken tikka masala (“Britain’s favourite dish”) is written टिक्का, transliterated as ṭikkā, IPA [ʈikkɑː].
The “hard” one appears to be a regional variant of the word usually written as tilak, the mark on the forehead of a Hindu.
In Nepal, Bihar and other regions, the tilak is called a tika (टिका)
…which in IPA is [ʈikɑː].
It would seem, then, that the difference is nothing to do with the t-sounds: both are retroflex and unaspirated. The difference is in the k-sounds. In one word the velar plosive is geminated; in the other, not.
I think that could have been more clearly expressed.