DAG Hamiltonian Path NP-complete The book computers and Intractability mentions that Hamiltonian Path problem is not NP-complete in DAG. 
But if Hamiltonian Cycle is NP-complete in digraph then I can split a vertex and create two vertices ${s,t}$ which will make the graph acyclic and then ask for $s\rightarrow t$ hamiltonian path in that DAG. These two problems are equivalent. Then why is one NP-complete and other is not ?
Your proposed approach doesn't work.  For some directed graphs there is no single vertex you can split to turn it into a DAG.  An algorithm only qualifies as a solution to the Hamiltonian Cycle problem if it solves the problem for all graphs.  It's not enough to solve the problem for a subset of "easy" graphs.