Usually there is little difference between truth and fact when closely examined. Here I present what I consider to be an example of the difference.
It is a fact that immigrants seek a better life in America because of problems in their native lands.
It is the truth that they should stay and work to solve those problems in their native lands.
Note that the truth statement contains a moral element.
Comments or other examples?
In philosophical logic, "truth" has no moral element. It is simply one of the two values a statement can assume (the other being "false").
A statement is a particular kind of sentence which is either true or false.
I would say that a fact is a true statement, but I don't know if this is its formal definition in philosophical logic.
An argument comprises a series of premises which the arguer maintains to be true and one or more conclusions that the arguer claims follow from the premises.
It seems to me that the examples you provide taken together comprise the following arguments (with one premise being an implied conditional statement):
Argument 1:
(P1) There are problems in immigrants' native land
(C1) Therefore, immigrants seek a better life in America
Argument 2:
(P2) Immigrants whose native lands have problems leave their native lands to seek a better life in America
(P3) If immigrants stayed in their native lands, then they would solve their native land's problems [IMPLIED]
(C2) Therefore, immigrants' native lands still have problems
For an argument to be valid, the conclusion must necessarily follow from the premise(s), regardless of whether the premises are actually true or not.
For an argument to be sound, the argument must be valid, and the premises must be true.
I would say that Argument 1 is not valid. That an immigrant seeks a better life in America is not necessarily because there are problems in the immigrant's country. For example, the potential immigrant's country may have no problems, but due to the scale of its economy the potential immigrant is only a millionaire, whereas he or she could become a billionaire if he or she immigrated to and worked in the United States.
I would say that Argument 2 is not sound. I think the conclusion follows from the premises, but (P3) is not necessarily true.