What is the difference between construction types mobile/manufactured and modular?

"My seller has a nice double-wide factory-built home on a full basement and does not want it to be listed as mobile/manufactured."

First, you may not misrepresent the property so you need to determine what it really is. All of these homes are intended for year-round occupancy in Wisconsin. They are not travel trailers, which are personal property, even if they are sitting on real estate (land).

Manufactured homes include mobile homes by Wisconsin statute 101.91(2)(c). They were called mobile homes prior to June 1976 and the HUD Title 6 standards. After that date they were required to conform to HUD standards regarding building codes for electrical, plumbing, heating, ventilation, fire protection, structural strength, etc. They should have a HUD tag attached to the home. The HUD Tag is a 2 x 4 inch aluminum plate affixed to the outside of the manufactured home. The HUD Data Plate is a sheet of paper inside, usually by the electrical panel, in a kitchen cabinet or bedroom closet and includes a map of the U.S. with information on wind zone, snow load, etc. that the home was built for.

Modular construction type is for homes built in a factory consisting of 2 or more modules that are assembled on site. These homes may be either “on-frame” or “off-frame” and will include a modular tag, usually under the sink. Modular homes must conform to local state building codes.

This is the MLS definition:
Manufactured/Mobile Home: Dwelling unit that conforms only to HUD codes rather than local building codes. A HUD tag should be found and this property is defined by Wis Stats., sec. 101.91(2) as: a structure that is designed to be used as a dwelling with or without a permanent foundation and that is certified by the federal department of housing and urban development as complying with the standards established under 42 USC 5401 to 5425 and includes any additions, annexes, foundations and appurtenances. A manufactured home sold on the land it sits on may be entered as a SFD, provided the construction is noted as manufactured/mobile home. A manufactured home is a minimum 400 square feet. A travel trailer (DOT- ANSI A119.5) is only personal property and may not be listed as a Manufactured/Mobile Home.

Modular: Home is constructed on site from modules. These homes must conform to local building code rather than Federal HUD codes. They are set on a permanent foundation or basement and do not have (never had) axles or a tongue for transport.

NADA provides a definition of the modular home built to local state codes and having a modular label.

The Wisconsin Housing Alliance also provides some information on determining the correct construction type of the home.