In learning a new language, when we are able to latch onto parallels with our own language, it can be comforting, but sometimes we have to let go and realize things work differently in the new one. That is the case with il passato prossimo. It has a name that makes us think that this tense is about a past that isn't very far away, because prossimo (with its cognate "proximate") does mean "near," "next," "close," etc.).
So when we learn that we use this tense to express things that have happened in the past and are already finished (as we use the simple past in English), it doesn't necessarily make sense. But let's look at it from another point of view. Let's look at it relatively. Because, although you can mostly get away with not using it, there is another past tense in Italian called il passato remoto. Here, too, we can detect the cognate remoto meaning "remote" or "far away." This is a simple tense in which the verb itself is conjugated. In general, it is used to express finished actions happening in the past that don't have any effect on the present. It means that there is a clear chronological and psychological distance between the fact expressed with the Passato Remoto and the present.
So compared to the passato remoto, the passato prossimo is closer, or less remote.
The passato remoto itself is not within the scope of this lesson, but let's mention that even when the passato remoto would be the preferred tense, we can usually get away with using the passato prossimo and lots of people do.
The passato prossimo is a compound tense that takes an auxiliary verb (avere or essere) and a past participle, but in a way, it is easier to use because we don't have to remember how to conjugate the verbs in the passato remoto. People will understand us and that's the most important thing. In addition to this, it's not always easy to know when to use the passato remoto. There are some grey areas.
The name "passato prossimo" refers to an action's place on a timeline. The name "present perfect," on the other hand, deals with the tense of the auxiliary verb we use ("to have" is used in the present tense in the present perfect). In the past perfect, the auxiliary verb is in the past tense. So the naming of the tenses has two different parameters, not to be compared.
The passato prossimo can express past actions that are over and done with (as the simple past does in English). But can also coincide with the present perfect in some instances.
To get an idea about when we use certain tenses, let's take a look at this video where two young women talk about their friendship. They talk about the past when they were in secondary school. They use the passato prossimo even though they are clearly talking about a time when they were younger.
E poi, dopo la maturità, abbiamo deciso di partire da sole con altri sei ragazzi di [sic: della] classe e siamo andati a Malta.
And then, after graduation, we decided to leave on our own with six other kids from the class and we went to Malta.
Captions 28-30, Erica e Martina La nostra amiciziaPlay Caption
As we mentioned above, sometimes the passato prossimo does coincide with the present perfect, as in this comment about their continuing friendship. Note that there is an adverb of frequency.
Ci siamo trovate sempre molto bene, in questi sei anni non abbiamo mai litigato.
We've always gotten along really nicely — over these six years we've never argued.
Captions 46-47, Erica e Martina La nostra amiciziaPlay Caption
But here, in the following example, they use the present tense to express what in English, we would express using the present perfect. Note the use of da (since, for).
Siamo amiche da sei anni,
We've been friends for six years.
Caption 3, Erica e Martina La nostra amiciziaPlay Caption
The above use of the present tense in Italian to express a "present perfect" situation is perhaps one of the trickiest tense differences to wrap our heads around. And it's just as tricky for Italians trying to speak English!
The two young women go on with the present perfect to talk about the past. Here, we find fa (ago), putting the action clearly in the past.
Ci siamo conosciute, appunto, sei anni fa, il primo giorno di scuola.
We met, in fact, six years ago, on the first day of school.
Captions 4-5, Erica e Martina La nostra amiciziaPlay Caption
If you are a subscriber it might be useful to watch the entire video to get a better feel for how the tenses are used. Looking at the transcript can help, too.