I found that sometimes 과의 is used instead of 과. For example,
The meaning seems to be the same however, in that it means "and" or "with".
So what is the reason you use 과의, instead of just 과 (and the same is true of 와의 vs 와 I think)?
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I am not 100% sure on the logic for this usage, but I will add my ideas about it.
A phrase including 와의/과의 ("●●와의/과의 ◆◆") requires that these two nouns come before and after 와의/과의, respectively:
a noun (or pronoun) referring to a living thing (usually, a person or a group of people)
a verbifiable noun referring to an action or a result of an action
a noun relating to another noun
a noun meaning a relationship, connection, difference, similarity, etc.
For Cases 1 and 2, it can simply be said as (1) [Doer]와의/과의 [Action/Result] and (2) [Target]와의/과의 [Action], respectively. The phrase for each case can be explained as follows:
Case 1a ([Doer]와의/과의 [Action]) shows that the doer does, did, had done, has done, will do, or is going to do the action together with another doer.
Case 1b ([Doer]와의/과의 [Result]) shows that the doer does, did, had done, has done, will do, or is going to do an action together with another doer to have the result.
Case 2 ([Target]와의/과의 [Action]) shows that a doer does, did, had done, has done, will do, or is going to do an action for/to/against the target.
These always imply that there is a doer omitted from the phrase (A word is omitted when it can be found in the context). I would say that the original construction for Case 1 is [Doer 1]와/과 [Doer 2]의 [Action or Result]. 의 has to be attached to the preceding word, so the ellipsis of [Doer 2] forms such a construction. This applies to your example, 대통령과의 인터뷰; you can take it as 대통령과 ■■의 인터뷰. The action noun is interview and one doer is the president (of the country). There is another doer (for example, journalists), but it is the context that determines who/what ■■ is.
Now, I think you have noticed a difference bewteen [Doer]와의/과의 [Action or Result] and [Doer]의 [Action or Result]. Although you can use either in some situations (for instance, when the action cannot be done by only one doer), the latter does not imply that there is another doer. Understandably, some action nouns, such as 독백 (monologue) and 독차지 (monopolizing), are unlikely to come after 와의/과의.
Similarly, the original construction for Case 2 seems to be [Target]과 [Doer]의 [Action]. The target does not do the action while the doer does. By doing the action, the doer affects or tries to affect the target. If you do not know what I am saying, please refer to the penultimate example that I will give far below.
There is one more requirement for Cases 1a and 2: When you convert the action noun into a verb, the verb needs to sound natural with "...와/과" for the phrase to make sense. Taking 친구와의 다툼 as an example, the verb 다투다 can be used with 친구와 as 친구와 다툰다, 친구와 다퉜다, etc.
Case 3 is different from the other cases because the word following 와/과 does not relate to an action. For this case only, I am sure that the original construction is [Noun 1]와/과 [Noun 2]의 [Noun 3]. When [Noun 2] is obvious, some writers omit it. One problem is that readers might not be able to find it. Examples for [Noun 3] are 관계 (Please note that it can be an action in a certain context and then it goes to Case 1), 차이, 차이점, 공통점, and 유사점.
In short, the construction ●●와의/과의 ◆◆ seems to have come from the construction ●●와/과 ■■의 ◆◆. This kind of logic will help you a lot to understand some other connections of postpositional particles.
As wilkvolk said, 대통령과 인터뷰 is a sentence fragment as it is missing the verb. It is also an incomplete phrase. This 과 does not mean and (as in A and B) because the two nouns are totally different in type or group. The noun 인터뷰 is verbifiable. If you use the verb 인터뷰하다, you should use phrases like 대통령과 인터뷰함, 대통령과 인터뷰하기, 대통령과 한 인터뷰, 대통령과 하는 인터뷰, and 대통령과 할 인터뷰 or sentences like "대통령과 인터뷰하였습니다," "대통령과 인터뷰합니다," "대통령과 인터뷰하고 있습니다," and "대통령과 인터뷰할 예정입니다." Please consider absolute and relative tenses when you choose one. You should also keep in mind that 인터뷰 may refer to its contents (내용) or something else relating to the interview(s). Then, you should use other phrases and sentences, for example, 대통령과 인터뷰한 내용 and 대통령과 인터뷰한 내용입니다.
When grammar does not really matter (like when you write a to-do list or text your plans to your friends), you can use such an incomplete phrase because you (and also your interlocutor) know what verb is omitted. For instance, you can just use 친구와 영화 관람 instead of 친구와 영화 관람하기 and 친구와 영화 보기 to mean "Watch a movie with my friend(s)."
These are other examples for 와의/과의:
너와의 추억 (When 추억 is the result of an action, another doer is usually "I." It refers to the memories "I" make, have made, made, am making, or will make with "you." If it is about the past, it means the memories I made with you (너와 쌓은 추억 or 너와 쌓았던 추억). Anyway, the literal translation will be "(The) memories with you.")
적과의 동침 (This is a translation of "Sleeping with the Enemy." It does not say who is sleeping with the enemy, but the translator could have used 적과 동침하기.)
친구와의 만남 (만나다 is a verb. 의 is unnecessary here.)
부모님과의 대화 (대화하다 is a verb; 대화를 나누다 is a collocation. It can be rewritten as 부모님과 나누는 대화, 부모님과 나누던 대화, 부모님과 나눌 대화, 부모님과 대화하기, 부모님과 대화함, etc.)
그대와의 노래 (노래하다 is a verb; 노래를 부르다 is a collocation. It can be rewritten as 그대와 부르는 노래, 그대와 부른 노래, 그대와 부를 노래, 그대와 노래하기, 그대와 노래함, etc.)
학교 폭력과의 전쟁 (It literally means a war with school violence, but you should understand it in a figurative sense. [Target] is school violence and [Action] is actively trying to prevent it rather than starting a war. The omitted doer will be teachers and other adults. It can be rewritten as 학교 폭력에 적극 대응함, 학교 폭력 근절 (노력), etc.)
그 사람과의 관계 (It can be interpreted in two ways. As the English word relationship does, 관계 has multiple meanings. If you follow Case 1, the phrase will mean a sexual relationship with the/that person because "●●와/과 관계하다" means to have a sexual relationship with ●●; if you follow Case 3, it will mean a relationship (not a sexual one) with the/that person.)
One more point that I must mention is that most people hardly use 와의/과의 in daily conversation except when there is 와의/과의 on/in something written that includes the lyrics of a song, the title of a movie, the title or contents of a novel, essay, or poem, and the contents of a letter and they need to mention it. We do not have to use such a construction even in our writing.