So LTP is generally thought to be mediated by an increase in the number of AMPA receptors at a synapse. There have also been studies showing that the size of spines is correlated with the number of AMPArs present at the synapse. Whether LTP actually induces the formation of more spines is not entirely clear. The general answer is 'we think the processes are probably related' but it is difficult to do the experiments needed to show that the in vivo correlates of LTP actually cause increased spine formation as part of their downstream signaling pathways. It is worth noting that most of our knowledge about LTP comes from studies in the hippocampus and that the relation between spine formation and LTP may vary from brain region to brain region and even from cell type to cell type.

tl;dr It is hypothesized that increased numbers of spines is a later stage of the process of memory consolidation that may initially appear as LTP however we need a whole lot more evidence to support those hypotheses.