The short answers: Yes and no; and yes we can.

The long answers:

From a chemical viewpoint, there really is no difference between plant and animal DNA because both comprise sequences of nucleotides in various orders. However, from a biological viewpoint, the order of arrangement of these nucleotides matters because it is what gives rise to genes and other genetic elements and these vary significantly between plants and animals (and is what gives rise to their biochemical, anatomical and physiological differences). Thus, because the set of genes (genes being arrangements of nucleotides in certain orders) and other genetic elements in plants and animals differ significantly, one can say that plant and animal DNA sets are relatively largely different, biologically speaking. 

However, there are certain genes common to both animals and plants. These are mostly some regulatory sequences that code for proteins that aid in processes like DNA replication, gene transcription/expression, post-transcriptional modification, RNA translation and post-translational modifications. Moreover, a general rule of thumb is that the more fundamental or basic a biochemical process is, the more likely it is that the genes behind the proteins controlling the process are similar in both animals and plants. 

As for transferring plant DNA into animals, yes it is possible and has probably been done. However, the more genetically distant organisms are from each other, the harder it is to successfully transfer genes or genetic sequences between them with desirable outcomes. Check out these bizarre genetic engineering transformations that may interest you :) : http://www.mnn.com/green-tech/research-innovations/photos/12-bizarre-examples-of-genetic-engineering/mad-science