What is the difference between a sound argument and a valid argument? In my notes, these are the definitions of a valid argument


  An argument form is valid if and only if whenever the premises are all true, then
  conclusion is true. An argument is valid if its argument form is valid.


For a sound argument,


  An argument is sound if and only if it is valid and all its premises are true.


Okay so to me, both definitions pretty much says the same thing to me. On a philosophy forum, I see that they distinguish the two by saying a valid argument is such that the truth value of the premises necessarily imply the truth values of the conclusion.

For example, the "Elimination" method say

$p \vee q$

$\sim q$

$\therefore p$

So the premises are $p \vee q$ and $\sim q$

Now if I were to substitute $p$ and $q$ for $p$ := "Jesse is my husband" and q:= "I am Jesse's wife" (assume p is true and q is true)

Then we have

Either "Jesse is my husband" or "I am Jesse's wife"

"I am not Jesse's wife"

Therefore, "Jesse is my husband"

So is this technically still valid or sound? (can't tell the difference) Both premises are true, but the conclusion is false? It should invalid right? Yet the method of elimination is said to be valid?
A sound argument is necessarily valid, but a valid argument need not be sound. The argument form that derives every $A$ is a $C$ from the premises every $A$ is a $B$ and every $B$ is a $C$, is valid, so every instance of it is a valid argument. Now take $A$ to be prime number, $B$ to be multiple of $4$, and $C$ to be even number. The argument is:


  If every prime number is a multiple of $4$, and every multiple of $4$ is an even number, then every prime number is even.


This argument is valid: it’s an instance of the valid argument form given above. It is not sound, however, because the first premise is false.

Your example is not a sound argument: $q$ is true, so the premise $\sim q$ is false. It is a valid argument, however, because for any $p$ and $q$, if $p\lor q$ and $\sim q$ are both true, then $p$ must indeed be true.

Note that an unsound argument may have a true or a false conclusion. Your unsound argument has a true conclusion, $p$ (Jesse is my husband); mine above has a false conclusion (every prime number is even).
Perhaps a valid argument is one that causes the listener to agree to some form of truth about a subject. A sound argument causes the listener to admit that there is more than one truth about the subject. It has the ability to raise the question of doubt. The difference lies in the listeners interpretation of the information provided. Arguments may be sound, but not necessarily valid. And valid arguments are not always sound. A good example of this is our judicial system....defense attorneys argument, prosecution attorneys argument and the jury or judges decision on the matter.