Complete ionic shows all of the ions even if they weren't changed in the reaction. Net only shows the ions that changed (precipitate, usually).
Eg. NaCl(aq) + Ag2+(aq)
complete: 2Na+(aq) + 2Cl-(aq) + Ag2+(aq) --> 2Na+(aq) + AgCl2(s)
net: 2Cl-(aq) + Ag2+(aq) --> AgCl2(s)
Eg. NaCl(aq) + Ag2+(aq)
complete: 2Na+(aq) + 2Cl-(aq) + Ag2+(aq) --> 2Na+(aq) + AgCl2(s)
net: 2Cl-(aq) + Ag2+(aq) --> AgCl2(s)