No, whilst they do collect on similar platforms, the difference is who is entitled to these royalties.
For example, publishing royalties are paid to writers (composition, lyrical content) and publishers, whilst Neighbouring Rights royalties are paid to those who own the sound recording and those who have performed on the recording.
In many cases, artists and labels may be entitled to both sides of the royalties, so it’s important to have a team who can go out and collect both.
Please note that each country around the world has their own appointed society paying publishing royalties and Neighbouring Rights, so having your publishing sorted does not mean that you have your Neighbouring Rights sorted too.