regarding DFT
Verification engineers will need a very thorough understanding of the functional operation of the chip, but does not care about anything beyond RTL design.
DFT engineer, on the other hand, needs to understand the physical architecture of the chip, the clocking speed and structure, the synthesis and timing aspects, and the manufacturing process. The DFT engineer's job starts very early in the project cycle, right when the architecture is being defined, and will last after the chip goes into production. The DFT engineer may be involved in yield improvement and customer returns.
The verification engineer's job typically ends when the chip goes into production, when all functional bugs are fixed.
So verification engineer will have more depth on functional operation of the chip, but the DFT engineeer will have more breadth on the whole process, from design to manufacturing to customer support.