Please consider the following strings array:
let strings = ["str1", "str2", "str10", "str20"]
Let's assume that what required is to get the first element (String) which contains 5 characters, I could get it by using filter(_:) as follows:
let filterString = strings.filter { $0.count == 5 }.first
print(filterString!) // str10
but after reviewing the first(where:) method, I recognized that I will be able to get the same output:
let firstWhereString = strings.first(where: { $0.count == 5 })
print(firstWhereString!) // str10
So what is the benefit of using one instead of the other? is it only about that the filter(_:) returns a sequence and the first(where:) returns a single element?
Update:
I noticed that the filter(_:) took 5 times to do such a process, while first(where:) took 4 times:

You are correct in observing that filter(_:) returns all elements that satisfy a predicate and that first(where:) returns the first element that satisfy a predicate. 
So, that leaves us with the more interesting question of what the difference is between elements.filter(predicate).first and 
elements.first(where: predicate).
As you've already noticed they both end up with the same result. The difference is in their "evaluation strategy". Calling:
elements.filter(predicate).first
will "eagerly" check the predicate against all elements to filter the full list of elements, and then pick the first element from the filterer list. By comparison, calling:
elements.first(where: predicate)
will "lazily" check the predicate against the elements until it finds one that satisfies the predicate, and then return that element.
As a third alternative, you can explicitly use "a view onto [the list] that provides lazy implementations of normally eager operations, such as map and filter":
elements.lazy.filter(predicate).first
This changes the evaluation strategy to be "lazy". In fact, it's so lazy that just calling elements.lazy.filter(predicate) won't check the predicate against any elements. Only when the first element is "eagerly" evaluated on this lazy view will it evaluate enough elements to return one result.
Separately from any technical differences between these alternatives, I'd say that you should use the one that most clearly describes your intentions. If you're looking for the first element that matches a criteria/predicate then first(where:) communicates that intent best. 
I believe we should start from considering each method separately and their purpose.
filter(_:) is not purposefully designed to prepare us to obtain first element. It is about filtering and that's it. It merely returns us a sequence. first can be used after filter, but that's just an instance of usage, a possible case. first is called for a collection and if we want, of course we are free to use it directly after filter.
But first(where:) was designed to filter and return a single element and is evidently the go-to approach to accomplish that kind of a task. That said, it is easy to presume that it's also preferred from performance perspective. As mentioned, we filter entire sequence, but with first(where:) we only process the portion of a sequence until condition is met.
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