What's the difference between Phage DNA and Recombinant DNA
Phage DNA is the DNA in a bacteriophage. 

Recombinant DNA is synthetically combined DNA used in genetic engineering.

In your case, since the phage's DNA is used as a vector, the phage DNA is the recombinant DNA, but that does not mean every phage's DNA is recombinant, since phages occur naturally. The point is, the phage DNA was altered to become recombinant DNA.