In the context of the HO model, comparative advantage is defined by the relative abundance of a production factor. Let's say country A has 10 units of labor and 10 units of capital, which country B has 8 units of labor and 4 units of capital. Country A has an absolute advantage in both labor and capital, but B has a comparative advantage in labor, because its labor to capital ratio (i.e., 2) is higher than country A's (1).
The HO model thus predicts that country A will export a good that needs relatively more capital for production to B, and that country B will export a good that needs relatively more capital for production to A.
Everything works beautifully in a world with two countries, two goods and two factors. But the world is nothing like that--it has many more countries, goods and factors than 2. Is this a big deal?
The whole point of economic modeling is to isolate the impact of a particular phenomenon, ceteris paribus. Ceteris paribus is a favorite phrase in the economist's lexicon, and means "all other things being equal or held constant." Sometimes asserting ceteris paribus is innocuous; often it is not.
A small change in the HO model creates serious problems. As Alan Deardorff showed, if the number of goods is greater than the number of factors of production, patterns of trade become indeterminate. The mechanics of the problem are simple: when the numbers of factors equal the number of goods, solving the pattern of trade problem involves equal numbers of equations and unknowns. This equality disappears when there are more goods than factors, we become unable to determine what is produced where. Allowing more goods than factors is not a trivial change to the model--it has an enormous impact on the analytical outcome. It is also a change that better reflects the reality of the world.
This is a fundamental problem with HO that is easiest to explain. Deardorff takes us through more sophisticated argues that show other problems with the predictions of the HO model. He expresses concerns that the model:
(1) implies fractions of good produced or trade routes utilized that are (unrealistically?) low;
(2) has a solution that is hypersensitive to [trade costs].
So the fact is we really don't have a theoretical model that predicts patterns of trade well. Yet we for years made lots of policy decisions based on a model that has lots of limitations.
[Update in response to comment: Krugman and Helpman are great in reconciling how interindustry trade happens and why countries with similar factors trade with each other. But the problems outlined by Deardorff about developing a robust general equilibrium model that predicts patterns of trade remain.]