What is a part of speech? What is a grammatical function?
Might as well ask: What's a noun? What's a verb? Why are they different?
This is the problem of the question. A noun is a person place or thing, and a verb is what a noun does or is.
Because a noun verbs, it doesn't stop being a noun as it verbs, so why should it stop being a noun as it adjectives?
Thus, why should a word defined as a specific part of speech by definition of what that part of speech is, change the part of speech because its function changes? The answer is: only if the definition of the noun is changed by changing the function.
What's a function? It's what a part of speech does.
You'll ask now, did I just verb the noun verb or did I use the noun verb erm. um... verbally?
Also, because it's not explicitly stated:
Where noun is used, part of speech can be reasonably interpreted to swap and the concepts hold. Where verb is used, grammatical function can be reasonably interpreted to swap and the concepts hold. Certainly not everywhere or else there will be an infinite loop and the Internet will implode.
Based upon comments:
baptism is a noun. A ritual is a noun. It's a thing being done.
So, when I say a verb is what a noun does, I mean a noun runs, walks, baptizes, screams, answers, shrugs, writes, types and any other word that describes, per TheFreeDictionary:
any of a large class of words in a language that serve to indicate the occurrence or performance of an action, the existence of a state or condition, etc.
Because the question keeps moving:
A verb is a part of speech that performs a grammatical function to indicate ... (the stuff quoted above).
Again the question moves:
OP in a comment that will be deleted soon enough:
So, according to that definition given by the free dictionary, (the kind of definition ridiculed by teachers and grammarians), arrival is a verb because it serve(s) to indicate the occurrence or performance of an action. You say that a verb performs a specific grammatical function, but you seem unwilling to give a simple definition of what a grammatical function is! I'm also confused because you seem to say that a noun is the doer of a particular verb. What about: The man was arrested. Is the man not a noun there?
a function from the Free Dictionary is:
The action or purpose for which a person or thing is suited or employed
Therefore, a grammatical function is the action or purpose for which a word is employed.
What are examples of grammatical functions?
cited within About.com
"The production and interpretation of an utterance act is anchored to the constitutive parts of language: syntax, morphology, phonology, semantics and pragmatics. While syntax is composed of structural units, for instance constituents in traditional grammar, phrases in functional grammar and generative grammar, groups in systemic functional grammar or constructions in construction grammar, it is the linear ordering of the individual parts within a hierarchically structured sequence which constitutes their grammatical function. The adverb really, for instance, realizes the grammatical function of a sentence adverbial with wide scope if positioned initially or finally, as is the case in the utterance really, Sarah is sweet. If the adverb really is positioned medially, it is assigned the grammatical function of the adverbial of subjunct with narrow scope, as in Sarah is really sweet. Or, the proper noun Mary can realize the grammatical function of object in Sally kissed Mary, and it can realize the grammatical function of subject in Mary kissed Sally. Thus, it is not the grammatical construction as such which is assigned a grammatical function. Rather, it is the positioning of a grammatical construction within a hierarchically structured sequence which assigns it a grammatical function."
(Anita Fetzer, "Contexts in Interaction: Relating Pragmatic Wastebaskets." What Is a Context?: Linguistic Approaches and Challenges, ed. by Rita Finkbeiner, Jörg Meibauer, and Petra B. Schumacher. John Benjamins, 2012)
Again from the OP in an ephemeral comment:
... or how about in They arrested the man? And regarding your concept verb: Are you saying that verb is a function? Or are you saying that verb is a part of speech? I'm confused.
I said a verb was a part of speech. It also performs a function. If you have to ask the question of whether a subject, direct object or an indirect object (all parts of speech called nouns) are in fact verbs because the action word (the part of speech called a verb) applies directly to the subject or one of the objects (nouns, whose grammatical function is a subject, direct object, or indirect object):
The man was arrested.
The: an article.
man: a noun. It's functioning as the subject. It doesn't change from being a noun because it's functioning as a subject. The definition of man does not change because of its grammatical function.
was arrested: a verb. It is functioning as the predicate. It doesn't change from being a verb because it has a function as a predicate. The definition of was arrested does not change because of its grammatical function.