I read somewhere that ancient Roman cuisine was closer to Vietnamese than to Italian cuisine. Garum (or fish sauce) was given as an example. Is this true and if yes then why did it change after the fall of Rome?
You should read Apicius, as it is a collection of Roman recipes. I can vouch mostly for the Italia part of Rome. The food is like Mediterranan and Italian cuisine minus vegetables and seasonings introduced after end of empire to the modern day.
As for De Re Coquinaria: Many meat recipes. Based on Pulse, Beans, vegetables and seasoning, it was closer to what we call "Mediterranean" food. Food in Italia changed after the Columbian Exchange and introduction of New World crops such as tomato and peppers. The spices and seasonings of Apicius can demonstrate the trade network with the East/India. I am fond of the spice 'Piper longum'. It is not like East Asian food at all. The vegetables are beets, leeks, lettuce, asparagus, cardone, escarole, turnips and are still used in regional dishes in Italian this day. There is also cooking with wine and herbs, which is also part of my families tradition and in modern cook books as well in Roman recipes. Do not get thrown off by the Garum. In my families and friends cooking, anchovie fillets were used to flavor food often, and this is seems in many recipes as an umami sort of flavoring.
Sometimes you have to see what Romans did not have to understand it better. Phaselus vulgaris Green beans, squash and pumpkin, eggplant(from the east), citrus with exception to citron, tomato, maize. Also look at modern Italian regional recipes and don't base you view on food of Italia on Olive Garden and the such.
You can see for yourself in this online version in English of the Roman Recipes.
http://penelope.uchicago.edu/Thayer/E/Roman/Texts/Apicius/home.html
Here is a page which is a companion to a PBS show which had a segment on Garum. This page has recipes with modern adaptations and explanation:
http://www.pbs.org/wgbh/nova/ancient/roman-recipes.html
Recipes and explanations with Latin and English from a book published by University Of Chicago Press:
http://www.press.uchicago.edu/Misc/Chicago/233472.html
I hope you enjoy learning and cooking the food of Ancient Rome to current day Mediterranean and Italia!