What is the difference between IAN and LNAV/VNAV?
A Non Precision Approach (NPA) flown with Integrated Approach Navigation (IAN) is not more precise than the same approach flown with LNAV/VNAV. The advantage of IAN is that the instrumentation and procedures are very similar to ILS (Cat I), which is the most common approach type flown for most airline pilots. Having a common set of indications on the PFD and procedures to follow improves flight safety.
The Boeing 737 Flight Crew Training Manual (FCTM) says:

IAN approach techniques are
similar to ILS approach techniques [...]
All IAN approaches provide the functions,
indications and alerting features similar to an ILS approach. Although non-ILS
approaches using LNAV and VNAV can still be performed, IAN is normally used
in place of LNAV and VNAV because of improved approach displays, alerts and
standardized procedures.

(Boeing 737 NG FCTM 5.60 - Approach and Missed Approach, emphasis mine)
The article about IAN on flaps2approach goes into a bit more detail:

Advantages of Using IAN
There are multiple advantages to an airline using IAN, the foremost
being flight safety. Unstable approaches contribute to many aircraft
accidents, and flight crews strive to always establish a stabilised
approach profile for all instrument and visual approaches.
The Global Position System has enabled stabilized approaches at many
airports and advanced features such as IAN take advantage of this
technology to provide consistent, intuitive displays that support
stabilized approaches.
18 Approaches Types to 1
Through the use of IAN, the number of approach types has been reduced
from 18 to 1, greatly simplifying the approach procedure and
minimizing the amount of time an airline needs to train pilots in
numerous approach types.  Time is money and utilizing advanced
technology such as IAN can increase airline productivity.
Additional Data - Increased Awareness
The distance to runway threshold, approach guidance information, and
vertical and lateral deviation markers are displayed when IAN is in
range of a designated runway.  Whether IAN is used or not, this
information provides additional guidance when executing an approach.
For example, when executing a VOR approach, this information has been
displayed on the Navigation Display (ND) as the distance to the actual
NAVAID (VOR) - which may or may not be aligned with the threshold of
the runway.  IAN will by default, display the lateral and vertical
deviation, and distance to the runway threshold, allowing for greater
precision during a non-automated approach.
These are but a few of the advantages to using the Integrated Approach
Navigation system.

(flaps2approach.com - Integrated Approach Navigation (IAN) - Review and Procedures)