The movie strongly implies this is because of Tauriel's status as a Wood Elf, or a Silvan Elf. Just like Tauriel, however, this prejudice seems to be entirely an invention of the movies.
Some simplified background from the book-verse on the types of Elves (you can detailed info here) - one of the ways Elves categorize themselves is by -
- the people that followed the Valar (gods/archangels) to Aman (something of a Paradise Island) - Galadriel is one of these, and Elrond, born in ME, is honorary one of these (via paternal lineage and upbringing)
- the people who agreed to follow the gods but for various reasons didn't actually end up completing the journey - Thranduil, Legolas and Tauriel.
- the people who they refused - nobody who appears in LotR or the Hobbit.
Now, among the second group of people, there was another split into 2. The first group gave up on the journey to Aman/Paradise when they came to the Misty Mountains. This is the group that later became known as Silvan Elves, ie. Tauriel
The second group got all the way to the coast, but then their leader / King got lost (long story) and they stayed behind looking for him. This was the group known as Sindarin Elves, ie. Thranduil.
Sindarin Elves, having undertaken more of the 'Holy journey' and having strayed out of duty, were 'higher' elves than Silvan elves, and by the time of the Hobbit there were few of them left in ME and those few were usually rulers over the much larger native populations of Silvan elves.
The movies seem to have taken this fact, and extrapolated a class/racial prejudice by the 'higher' Sindar towards the 'lowly' Silvans, which is why Tauriel isn't good enough for Legolas.