The difference between them is their downstream effects. When ligand gated ion channels are opened up by an appropriate ligand (endogenous or exogenous drug) they'll let in ions for a period of time. These can be either positively charged or negatively charged, such as Ca2+ or Cl-. Normally a cell will has an inner membrane resting potential of around -70mV, when more positive ions are brought into the cell it's charge will become more positive (depolarize), making it more likely to fire an action potential. Negative charged ions are going to make the cell more negative (hyperpolarized), making it less likely to fire (inhibition).
GPCRs on the other hand work through signalling cascades. When a ligand binds to a GPCR, depending on the subunits, its going to activate different messenger systems in the cell, which can all alter cell function. For example, increasing/decreasing transcription of a gene.