As others have pointed out, two genes are homologs if they evolved from a common ancestor. Two genes are orthologs of each other if, in the gene family tree, the last common ancestor of those two genes was a speciation event. Two genes are paralogs of each other if, in the gene family tree, the last common ancestor of those two genes was a duplication event. This is important to know because after duplication, the genes are more likely to separately evolve more specific functions (subfunctionalization) or even new functions (neofunctionalization).
Berkeley PHOG provides a precomputed database of orthologs derived from PhyloFacts gene family trees using tree distances. The default, most stringent variant, PHOG-S, aims to predict only clusters of genes which are all superorthologs of each other, that is, there are no duplication events in the portion of the gene family tree containing them. This variant has high precision but relatively lower sensitivity. The thresholded variant, PHOG-T, ignores putative duplication events that are very close to the leaves of the gene family tree (i.e., very recent in evolutionary history). How close is "very close" is a tunable threshold which you can set based on the taxonomic distance you're interested in. E.g., the "Close" threshold (available as a preset) was tuned for human-mouse orthologs. You can change the threshold on the fly and get more and more predicted orthologs of your query sequence. The ones you got first (at a lower threshold) are closer (by tree distance, and with less duplication events in between) than the ones you got later (after increasing the threshold). This suggests they may be closer in function as well, providing an answer to your original question.
For example, this search for superorthologs of ALG2_MOUSE provides a single human superortholog, ALG2_HUMAN. But searching at the close threshold brings up another closely related human sequence, Q8NBW5 (a clone sequence that might actually have differed from ALG2_HUMAN only due to sequencing errors). The sequence ALG2_HUMAN found at the superortholog threshold (i.e., threshold 0) is closer to ALG2_MOUSE than the sequence Q8NBW5 found at the higher, "close" threshold.
The Berkeley PHOG ortholog report is also available as CSV, e.g., orthologs for ALG2_MOUSE in csv format. You can see how to construct the url if you would like to do so programmatically. We have a student working on providing them in OrthoXML format as well.
Hope this helps. Please let me know if you have further questions!