Words
The main Tamil vocabulary is based on the Dravidian culture. There are a few influences by other cultures like the Malays, Chinese or even Greeks that did trade with Tamil kingdoms but Tamil purism kept the language from having lots of foreign words in Tamil.
Also, Tamil is, compared to other Dravidian languages, not that much influenced by Sanskrit. Only the wish to pick up the Indo-Aryians religion Hinduism as the main Tamil religion made the religious texts and some abstract words being in Sanskrit.
While Tamil was influenced by other cultures and languages, Tamil itself also influenced others. The English word orange is probably the best example that you can trace back to a Tamil origin: நார்த்தங்காய்「nārttaṅkāy」
Another, more known example, is the word rice, coming from the Tamil word அரிசி「arici」. However, the path of this word can't be proven properly.
While you can split Tamil in three different stages "Old Tamil", "Middle Tamil" and "Modern Tamil", Tamil demonstrates grammatical continuity across these stages. Many things that you will learn in this book have their origin in Old Tamil.
Modern Tamil language is a language with a difference between the written and spoken language. While the spoken language derives from the written one, there are a few dialects that were influenced more or less by their surrounding neighbors.
While in Tamil Nadu English has quite a big influence on the spoken Tamil, the Tamil spoken in Sri Lanka is close to written Modern Tamil. But even there you will notice a different dialect between Jaffna Tamil and Batticaloa Tamil.
In this book you won't learn any of the dialects but the Tamil Grammar of written Tamil. This should, though, not hold you back from using this grammar for spoken Tamil as well. People might even be impressed by that!