There are not synonymous, actually the diference is quite simple:
Sudamérica: Geographically, a subcontinent of "América", limited to the north by the Caribean sea. It does not include the Caribean, Central America ("Centroamérica") or North America ("Norteamérica"). Argentina, Brazil and Suriname are in "Sudamérica", but not Costa Rica, México or Cuba.
Latinoamérica: All american countries where a romance language (Spanish, Portuguese, French) is spoken as main language. Includes countries of South, North, Central America and the Caribean. Argentina, Brazil, Costa Rica, México and Cuba are in "Latinoamérica", but not Suriname. Particularly, Canada as country is not included despite French is spoken in some provinces.
Hispanoamérica: All american countries where Spanish is spoken as main language. Includes countries of South, North, Central America and the Caribean. Argentina, Costa Rica, México and Cuba are in "Hispanoamérica" but not Brazil or Suriname.
It's important to remark that there is no term to refer "all countries to the south of US", and the terms described before can't be used for that purpose.
"Latinoamérica", "Hispanoamérica" and "Iberoamérica" have cultural and linguistical implications, so are more complex to use. I would suggest to use "South America", "Central America" and "México" (only latin american country in North America) acording to the case, but I guess other people could think different.