LTTE has always had (and still has) very very DEEP and Extensive support base not only in S. Lanka, but in India, UK, USA, and Canada. Tamils just like Talibs were the local guys well entrenched in the S. Lankan society for centuries if not longer.
Your correction is incorrect :)
LTTE has had support outside SL (including India), but not within the common Sinhala public in SL or the SL government apparatus. Al Qaeda/Taliba on the other hand have support of sections of the Pakistani public and Pakistani state. That is the reason there was/is a reluctance to put down Taliban with an iron hand.