What's the difference between "Nonstop Commercial Air Tour" and "Aerial Photography"?
Short answer: as far as I can tell, an air tour implies photography, but photography doesn't imply an air tour. The regulations define various criteria for determining if a flight is an air tour, but there's no 100% clear definition.



First, the regulatory definition of "commercial air tour" is in 14 CFR 110.2:


  Commercial air tour means a flight conducted for compensation or hire in an airplane or helicopter where a purpose of the flight is
  sightseeing. The FAA may consider the following factors in determining
  whether a flight is a commercial air tour: [...]


It's worth reading the "following factors", by the way.

Second, there's no formal definition of "aerial photography". See the Sapp (2007) interpretation (emphasis in the original):


  In a 1989 interpretation, the FAA discussed the meaning of "aerial
  photography" [...]
  the phrase was not defined in the regulations [...]
  "aerial photography" connotes a condition where taking pictures or
  filming is done from the air


Third, the Ragland (2015) interpretation specifically mentions photography as one purpose of an air tour (emphasis mine):


  Specifically, you asked whether §119.l(e)(4) allows persons to conduct
  commercial air tour operations, for purposes of sightseeing and
  photography, without obtaining either a letter of authorization
  (LOA) in accordance with §91.14 7 or a commercial operating
  certificate in accordance with part 119 of the federal aviation
  regulations.  
  
  You assert that the conclusion in a January 5, 2011
  interpretation issued by our office "appears to reinforce the need for
  Operators to apply for a LOA from their local FSDO to conduct
  'sight-seeing tours' within a 25 statute mile limitation under 14 CPR
  91.147." We agree.


Finally, the same interpretation says you can't operate a commercial air tour under the aerial photography exemption:


  We further note that the exception in §119.1(e)(4) for certain "aerial
  work operations," such as banner towing, aerial photography or survey,
  and powerline or pipeline patrol, does not extend to air tour
  operations in which the primary purpose is sightseeing.


So, putting all that together - and remembering that I'm just SGOTI - it appears that:


Air tours within 25SM require an LOA and are assumed to include photography
"Aerial photography" is a separate activity from "sightseeing"
The FAA seems interested in the "primary purpose" of the flight


In your scenario, it sounds like the photographer is asking for a straightforward aerial photography flight: the primary purpose is to take photographs, not to get a tour of an area. And if the photographer actually makes money from their work, that's a strong indicator that they aren't just "sightseeing". But that's just my opinion.