The topology of the tree is determined by the tree file, it just looks different due to where you put the root of the tree and how taxa are ordered (tree topology is rotation invariant). In other words, the topology is identical but branches can be rotated. In both applications you should be able to do the following:

• Select the outgroup to root the tree
• If you don't have a good outgroup, midpoint root the tree (command-M in FigTree)
• Then sort the tree in increasing or decreasing node order (command-U)

Then you will find that the topology is exactly identical. I think, if you do this once in FigTree and save it as a Nexus format file, then load this file into itol, the ordering should remain the same (not sure though).