Wa Alaikum As Salam.
Ahl Al Sunnah and Ahl Al Hadith are both synonymous descriptions that describe Muslims. They are just descriptions.. indicating that these people follow the Sunnah or follow the Hadith. If you are a Muslim with correct Aqeedah & following Islam properly as taught by the Prophet Muhammad (ﷺ), then either of these terms can apply to you.
IF there is any difference, then it is in that 'Ahl Al Hadith' studied hadith more extensively. But again, you don't have to be an hadith expert to fall into this category. Any Muslim who accepts hadith and follows them sincerely is of this group.
Note: Many classical scholars have commented on 'Ahl Al Hadith'. It's not a new term!!
These terms have been used to distinguish ourselves from those who claim to be Muslim yet don't follow Sunnah/Hadith and engage in innovations. Does this contradict the idea of 'just be Muslim'/'don't form sects'? No. Again, they are just description of what it means to be Muslim. - There is no problem in a description. It is of Islam to disassociate yourself from evildoers.
As for Dr. Zakir Naik, he is not a scholar but he's good in comparative religion. Some people from the Ahl Al Hadith movement especially in India may/do respect him a lot, but that doesn't make him the 'face' of Ahl Al Hadith in general. (In other words, whether you follow him or don't follow him doesn't affect your standing). He personally wears a suit and cap, but the hasn't told anyone else to imitate him nor should anyone else go to great lengths to do so.