What is the difference between Boeing 73G, 73H and 737?
There are two main systems of aircraft type designations: IATA and ICAO codes. The ones you mention are IATA codes (all 3-character) and are often used for shortened aircraft type designation in public information.

Both code types are listed here with the corresponding full aircraft types. As you can see, the IATA information covers the type and variant, including some features such as winglets, but not necessarily extended range models or sub-variants with different engines. It also differentiates between freighters and passenger versions. 

Despite the additional digit, ICAO codes are less precise and often include several variants such as freighter as well as passenger variants of the same type.

The codes given in your question translate to:


733  - Boeing 737-300
734  - Boeing 737-400
735  - Boeing 737-500
738  - Boeing 737-800
73G  - Boeing 737-700 (7th letter of the alphabet)
73H  - Boeing 737-800 with winglets (8th letter of the alphabet)
762  - Boeing 767-200
763  - Boeing 767-300
772  - Boeing 777-200
773  - Boeing 777-300
777  - ambiguous 777 type


(all passenger aircraft)

Unfortunately, as @Ralph J mentions, it's not always clear which type of code is used. Mostly, a 4-digit-code will be an ICAO code and a 3-character-code will use the IATA system, but because some ICAO codes only contain 3 digits, this isn't always reliable. For example, if you read "DC6", it could be the IATA code and refer to the passenger version, or it could be the ICAO code and you wouldn't know if it refers to the passenger or the freighter version.