The protostomes are a monophyletic group, i.e. they include all the descendants of some animal that lived a short time before the beginning of Cambrian, probably around 550 million years ago.
On the other hand the deuterostomes might be or might not be a monophyletic group, it is not known yet with certainty, so it is still possible that the word "deuterostomes" might be invalid, if there exists no such group.
The name "deuterostomes" is used for the ensemble of 2 groups, the chordates (including us) and a group composed of echinoderms and hemichordates (Ambulacralia).
These 2 groups share one character present in their ancestors, the existence of some openings that connect the pharynx with the exterior (e.g. the ears in humans or the gills in fish).
However, there is a chance that this character also existed in the ancestor of the protostomes, but it was then lost, in which case it would not be a proof of a closer relation between chordates and echinoderms+hemichordates.
The ancestor of the protostomes and the ancestor of the chordates both had a rather complex nervous system, while the ancestor of the echinoderms+hemichordates had a much simpler nervous system, similar to that of many of the cnidarians (but even some jellyfish have a more complex nervous system than that).
If it will be proven in the future that in fact the protostomes are more closely related to the chordates than the echinoderms(+hemichordates), then their common ancestor would already have had a complex nervous system.
In this case, the similarities between octopuses and mammals, while still greatly increased by convergent evolution, would be far less surprising, because there would be much more features of the nervous system that would be inherited from the common ancestor.
While there are many genetic similarities between echinoderms+hemichordates and chordates (i.e. us), most of those are also present in cnidarians, so they must be older than the separation between chordates and protostomes. The chordates retained them but the protostomes lost those features, so they are not a definitive proof of relationship.
Which is the closest relative of the chordates is one of the most important questions of the animal phylogeny and there are chances that the answer that was believed true for more than a century is in fact wrong.
The correct answer to that question has consequences mainly for understanding the evolution of the nervous system and for the applicability of some things that we can learn about the brains of protostomes (like the octopuses) to our own brain.
On the other hand the deuterostomes might be or might not be a monophyletic group, it is not known yet with certainty, so it is still possible that the word "deuterostomes" might be invalid, if there exists no such group.
The name "deuterostomes" is used for the ensemble of 2 groups, the chordates (including us) and a group composed of echinoderms and hemichordates (Ambulacralia).
These 2 groups share one character present in their ancestors, the existence of some openings that connect the pharynx with the exterior (e.g. the ears in humans or the gills in fish).
However, there is a chance that this character also existed in the ancestor of the protostomes, but it was then lost, in which case it would not be a proof of a closer relation between chordates and echinoderms+hemichordates.
The ancestor of the protostomes and the ancestor of the chordates both had a rather complex nervous system, while the ancestor of the echinoderms+hemichordates had a much simpler nervous system, similar to that of many of the cnidarians (but even some jellyfish have a more complex nervous system than that).
If it will be proven in the future that in fact the protostomes are more closely related to the chordates than the echinoderms(+hemichordates), then their common ancestor would already have had a complex nervous system.
In this case, the similarities between octopuses and mammals, while still greatly increased by convergent evolution, would be far less surprising, because there would be much more features of the nervous system that would be inherited from the common ancestor.
While there are many genetic similarities between echinoderms+hemichordates and chordates (i.e. us), most of those are also present in cnidarians, so they must be older than the separation between chordates and protostomes. The chordates retained them but the protostomes lost those features, so they are not a definitive proof of relationship.
Which is the closest relative of the chordates is one of the most important questions of the animal phylogeny and there are chances that the answer that was believed true for more than a century is in fact wrong.
The correct answer to that question has consequences mainly for understanding the evolution of the nervous system and for the applicability of some things that we can learn about the brains of protostomes (like the octopuses) to our own brain.