Is it true that the possessive form is not used for a noun which is being compared to by "that of" or "those of"? For example:
... making your products or services different from and more attractive than those of your competitors'.
Also, could I use the possessive without using "that of" or "those of"? For example:
Lincoln's party is better placed to win the election than Roosevelt's.
Yes, it is true — if, by "possessive form" you mean the possessive inflection of a noun ('s, etc.), and yes, you can use the possessive without using that of or those of. However, I don't believe you fully understand the two ways to indicate the possessive case. I'll explain them, and then explain how the demonstrative pronoun (this, that, these and those) fits in.
The possessive case takes either of two forms: inflection of the noun, and including the noun in a prepositional phrase with of. These two sentences are equivalent in meaning:
My father's birthplace was New York.
The birthplace of my father was New York.
The difference is a matter of style and clarity. For example, your first sentence could be worded in this way:
... making your products or services different from and more attractive than your competitors'.
The entire meaning of the sentence is changed with the single apostrophe! If it weren't there, the products or services would be more attractive than the competitors themselves, rather than their products or services. So, the sentence as you have it is much more clear (although you need to remove the apostrophe if you use those of — do one or the other).
Likewise, your second sentence could be worded in this way:
Lincoln's party is better placed to win the election than the party of Roosevelt.
This sounds pompous, as it adds to the complexity of the sentence without clarifying its meaning.
Now demonstrative pronouns are incidental to the above. A demonstrative pronoun, like any other pronoun, simply takes the place of a noun. Let's look at four ways to say your second sentence:
Lincoln's party is better placed to win the election than Roosevelt's party.
Lincoln's party is better placed to win the election than Roosevelt's.
Lincoln's party is better placed to win the election than the party of Roosevelt.
Lincoln's party is better placed to win the election than that of Roosevelt.
The first two sentences demonstrate the ability to remove the noun party when it's understood by context. The second two sentences show the substitution of the party with the demonstrative pronoun that. Since both sentences are equally in the possessive, that substitution isn't a part of what makes the possessive case.
So, you may always change the possessive inflection to the prepositional phrase with of and vice versa. Often, but not always, you will accompany the prepositional phrase with the demonstrative pronoun, since the noun to which it refers is often given elsewhere in the sentence. But the demonstrative pronoun is not a part of the rule in question.
To further show this, the reason that this is not correct:
Lincoln's party is better placed to win the election than that of Roosevelt's.
Is the same reason that this is not correct:
Lincoln's party is better placed to win the election than the party of Roosevelt's.
Both of them are mixing the two different forms of the possessive. You have to use one or the other.
