Why might a large predator and a small animal that it eats have different adaptations?

Here is one reason:

A large predator may have little to no predators.  The small animal (which is eaten by the particular large predator may have different adaptations.  For instance, the small animal is at risk for being eaten by the large predator—and potentially other predators, too.  Some strategies might include mimicry, playing dead, camouflaging (e.g. blending in with blending in with vegetation),  producing in larger numbers (for survival of the organism—being "r-strategists".
   The large predator would tend to be a "k-strategies".  The large predator would not have to worry as much about be hunted for prey.  The large predator may have specific characteristics that differ from those of the small animal (which the said large predator eats) and many need different strategies (e.g. the large predator may be a reptile and need thermoregulation/ sensitive to temperature/ seasonal changes;  may change skin/coat color to absorb/reflect heat.

Proteins are the workhorses of the cell, controlling virtually every reaction within as well as providing structure and serving as signals to other cells. proteins are long chains of  amino acids  , and the exact sequence of the amino acids determines the final structure and function of the protein. instructions for that sequence are encoded in  genes  . to make a particular protein, a messenger ribonucleic acid (mrna) copy is made from the gene (in the process called  transcription  ), and the mrna is transported to the  ribosome  . protein synthesis, also called  translation  , begins when the two ribosomal subunits link onto the mrna. this step, called initiation, is followed by elongation, in which successive amino acids are added to the growing chain, brought in by transfer rnas (trnas). in this step, the ribosome reads the  nucleotides  of mrna three by three, in units called  codons  , and matches each to three nucleotides on the trna, called the anticodon. finally, during termination, the ribosome unbinds from the mrna, and the  amino acid  chain goes on to be processed and folded to make the final, functional protein