What are the differences between Sri Lankan Tamil and Indian Tamil diaspora?

 Differences are many.

1. As Tamil is a diglossic language the differences between the standard written languages across the globe is minimal but the spoken varieties differ considerably.
2. In the sangam age in Tamil Nadu, the classical form of Tamil is classified into three - Iyal (literature), Isai (music) and Nadagam(theatre). Tamil, in any point in Time and Space, has borrowed the influence of these three in varying propositions. You can see how the dialogues of Tamil films in Sivaji Ganesan era (1970s) and now (2018) changed. So as the Tamil words written in India and a distant Sri Lanka. However these differences are minimal in comparison with spoken language.
3. Coming to dialects - leaving aside, Madurai Tamil, Kongu Tamil, Madras Tamil! - Sri Lankan Tamil itself has four dialects. i)Jaffna Tamil - close to Tamil Nadu Tamil II)Batticaloa Tamil is more literary and unique Tamil iii)Negomba Tamil - Fishermen Tamil - bilingual, strongly influenced by sinhala iv)Moors Tamil - Spoken by Moors strongly influenced by Arabic
4. The spoken varieties in Sri Lanka although different from those of Tamil Nadu share some common features between the southern dialects of Tamil Nadu and the northern Sri Lankan Tamil dialect, the language of Jaffna. Sri Lankan Tamil dialects retain many words and grammatical forms that are not in everyday use in India, and use many other words slightly differently.
5. Sri Lankan Tamil dialects are less influenced by Sanskrit and English.
6. Sri Lankan Tamil dialects are inflenced by Portugese, Dutch as well as English and Sinhalese.
7. In general, Sri Lankan Tamil is considered to be more conservative than continental Tamil.