Is there any differences between Indian Tamil and Sri Lankan Tamil?

 Differences are many.

1. Written and Spoken: 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. Iyal, Isai and Nadagam: 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. Indian Tamil Dialects: Coming to dialects - Tamil spoken in India differs every 250 kilometres - Kumari Tamil, Madurai Tamil, Chetti Nadu Tamil, Kongu Tamil, Madras Tamil and many.
4. Sri Lankan Tamil Dialects: It 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
5. Commonality: The spoken varieties in Sri Lanka although different from those of Tamil Nadu share some common features between the Southern dialects (Kumari, Madurai) 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.
6. Influencers of Indian Tamil: Indian Tamil dialects are influenced by Sanskrit and English.
7. Influencers of Sri Lankan Tamil: Sri Lankan Tamil dialects are inflenced by Portugese, Dutch as well as English and Sinhalese.
8. Conservative: In general, Sri Lankan Tamil is considered to be more conservative than continental Tamil.

Migration History of Sri Lankan Tamil:

Languages get diluted with space and time. But descendents who migrated maintain the purity of the time in which they migrated. That is why it is importnt to understand the migration history. It is outside the purview of the question and given as an information.

#The migration consists broadly in two stages -who arrived in 7th to 10th centuries ago, during Chola Rulers to Anuradhapura mainly soldiers; and during British in 19th century as a trade workers.

Foot Note - The outward Indian migration on an organized scale came during British Rule. The commissioning of railway line in 1860 made the trading easy across India. In overseas, Parsis running into China; Gujaratis into Red Sea, Persian Gulf; Sindworkies into Ports in Americas, Europe. For the Nattukottai Chettiars of Tamil Nadu, the chief investment destination was South East Asia. From around 1830, confined Chettiars in Madras presidency, started exporting their capital to Ceylon(Sri Lanka), Burma (Myanmar), Straits settlement (Singapore, Malacca and Penang) and Indo-China. As per the Indian Estimate, the working funds of Chettiars were deployed 64% in Burma, 20% in Malaya, 11% in Ceylon, 4% in Indo China. Less than 1% is deployed in Madras. Chettiars taken the plantation workers all over the places.