A phylogenetic tree is a tree showing relationship between lineages. These lineages might be computed for genome-wide DNA or from only a single gene. As such the term phylogenetic tree is general.
gene tree vs species tree
If you compute a phylogenetic tree, from genome-wide DNA, then you are computing a species tree (although some sister lineages might not perfectly fit the definition of species). If the phylogenetic tree is computed from data coming from a single gene, then we are talking of gene tree.
Why would a gene tree not match a species tree?
But a gene tree is not just a species tree with fewer data. There are reasons for why a gene tree may not match a gene tree. Genes may duplicate within a given genome. The two duplicated genes are free to evolve independently since the split. All descendants lineages will then inherit these two genes but if you map the two different copies of the gene (red and green below) in two sister clade, they will be much more different then the two same copies (say the green copy) in two distantly related species. Here is a picture to show that
Also, copies of genes can be deleted later on and there might have horizontal gene transfer. For all these reasons the phylogenetic tree of a gene might well be very different than the phylogenetic tree of species.
Below is a gene tree and the associated species at the tips. Take time to make sense of what is going on on the picture.
What is a species tree exactly?
One could call a species tree as some kind of average tree among all gene trees.