Tatsam: In the simplest terms it is the use of Sanskrit word in their more original form.
Tadbhav: These words may have been derived from Sanskrit but their form has been morphed overtime (depending on local accents, to make them easier to speak by common people)Explaination: Tatsam are Sanskrit loanwords in modern Indian languages esp. (Bengali, Marathi, Oriya, Hindi, Gujarati, Sinhala and also Dravidian languages like Malayalam, Kannada and Telugu. They are more erudite than common words. This can be compared to using words of Greek origin in English (e.g. hubris),
It is Important to note here that while all of the North Indian languages descend from Sanskrit, Dravidian ones (the ones spoken in South India) do not have Sanskrit origin still they borrowed heavily from the same. This is not analogous to English and Greek / Latin.
A Tatasam can have multiple Tadbhav forms depending upon the local language. Sometimes with the morphing of the word even the meaning gets somewhat customized. This was done to fill in the gaps in a language.
History: Tatsam originated in tenth century poetry, when the poets felt that the colloquial language was not suitable for their expressive needs. Sanskrit went to the Dravid land mainly because of the pioneering work of Nannaya, who translated Mahabharat in Telugu. Currently used Telugu consists of 60% Sanskrit based words (Tatsam + Tadbhav ). Even the metrical poetry of many Dravid languages use pure sanskrit meters.