Solution

Weigh tiles 1-4 against tiles 5-8. (weighing no. 1)

Case 1.
Tiles 1-4 weigh the same as tiles 5-8.
Therefore, the different tile is 9, 10, 11 or 12.
Now combine tiles 9, 10, 11 and 1 and weigh against 2 through 5. (weighing no. 2)

Case 1a.
If the two piles weigh the same, tile 12 is the different tile.
Weigh tile 12 against tile 1 to determine whether tile 12 is lighter or heavier than the rest. (weighing no. 3)

Case 1b.
If the pile of tiles numbered 9, 10, 11 and 1 is heavier than 2-5, then you have deduced that the different tile is either 9, 10 or 11 and is heavier than the rest. If the pile is lighter, then it follows that the different tile is lighter. Knowing whether the different tile is lighter or heavier makes it simple to determine which of the three is different: weigh 9 against 10. If they are the same, tile 11 is different.. (weighing no. 3)

Case 2.
Tiles 1-4 are heavier than tiles 5-8. We currently do not know whether the different tile is 1-4 and is a heavier tile or whether it is 5-8 and is a lighter tile.
The next step is to remove tiles 1, 2 and 3 on the left side and replace them with 5, 6 and 7. On the right side we add in tiles 9, 10 and 11 which we know to be normal tiles now. Reweigh. (weighing no. 2)

Case 2a.
If the two piles weigh the same, either tile 1, 2 or 3 is different and is heavier than the rest. Weigh tile 1 against 2. If they are the same, then tile 3 is different and is heavier. (weighing 3)

Case 2b.
If the left side (tiles 4, 5, 6, 7) is still heavier than the right side (8, 9, 10, 11), we know that either tile 4 is different and is heavier or tile 8 is different and is lighter. Weigh tile 4 against tile 12. If they are the same, tile 8 is different and is heavier, else tile 4 is different and lighter. (weighing 3)

Case 2c.
The left side (tiles 4, 5, 6, 7) is now lighter than the right side (8, 9, 10, 11). Therefore we deduce that tiles 4 and 8 are normal because they have not been moved. Additionally on the right side, tiles 9-11 are normal which means that all the tiles on the right side are normal. Either tiles 5, 6 or 7 are different and are lighter than normal. Weigh tile 5 against tile 6. If they are the same tile 7 is different and lighter. (weighing 3)

Case 3
This is the opposite scenario of Case 2 where tiles 1-4 are lighter than tiles 5-8. The same logic applies to discover the different tile.