Q:

Why use quote tags when quotation marks will do?

I'm new to coding and have a trivial question: Why is it necessary to use the quote tag when it's easier to just write out the quotation marks in the text?

A:

It's All About Semantics
The purpose of HTML is to allow you to add semantic information to the resource. In other words, when you surround a quote with quote tags, you are describing to the program that will be using this resource as to what the content means. Programs aren't always just browsers that render the HTML into an image for us to view; they might be screen readers for the vision impaired or a program that reads information from a web page and inserts data into a database (such as the web crawler for a search engine).
Why Semantics?
A similar question to yours would be, "why use a header tag when I could use a  tag with a custom style to make the font larger and bold?"
The reason is because by marking text with a header tag (h1, h2, etc.) you are telling the program reading the HTML document that the content has special meaning. The program can then do things with the document besides simply displaying it to the user; if the HTML document has header tags in place, the program could automatically create a table of contents of the document by simply listing out the contents of the header tags (similar to how a Wikipedia article can automatically create a table of contents on the top of the page).
So, everything starts with adding semantic information. As others have pointed out, you can style the content of a quote tag, where you cannot style content within two quote characters. This is a by-product of adding semantics, however, and not necessarily the end goal. Of course, you could have styled the quotes by surrounding it with <div class="quote">..</div>. However, by doing this you lose the ability to have the browser help you render the quote as you like (see the quotes css attribute), or even have the browser render the quotation marks using the default quotation marks of the user's locale.
Even after this, there is more to the quote element than styling. For example, the quote element offers the ability to show additional information with a "site" attribute.
How could semantic information be used for quotes?
For example, let's say I create an HTML page called "http://example.com/MyThoughts". In that page, I have the following HTML...
<p>The W3C page <cite>About W3C</cite> says the W3C's mission is <q
cite="http://www.w3.org/Consortium/">To lead the World Wide Web to its
full potential by developing protocols and guidelines that ensure
long-term growth for the Web</q>. I disagree with this mission.</p>

Notice the cite attribute on the <q> element here (not to be confused with the <cite> element). If people added quote elements in this way, we could now create a web crawler that goes through the internet looking for pages that have a quote and citation. Then, using that data, we could create a database of documents and their citations. We could create a new site where the output might look like...
Resource: "About W3C"
Location: http://www.w3.org/Consurtium
Document Citing This Resources:
    Resource: "My Thoughts"
    Location: http://example.com/MyThoughts

...or equally useful...
Resource: "My Thoughts"
Location: http://example.com/MyThoughts
Documents Cited in this Resource:
    Resource: "About W3C"
    Location: http://www.w3.org/Consurtium

As you can see, we have created an application that joins together data from other websites without the need for APIs or direct database access. That is the power of adding semantics to your documents.
Conclusion
When doing day to day development work, the possibilities of what interesting things can be done be introducing semantics into your HTML documents is typically ignored in place of ensuring that the site "looks good in a browser". Semantically adding information can still help due to things such as helping you style your quotes correctly for the specific user.
To be sure, if you used a quote character instead of a quote element Tim Berners-Lee is not going to come bust down your door. However, the built-in browser rendering of quotes specific to the user's locale is a nice carrot on the stick.

A:

It certainly is not necessary, but there are a number of reasons why using <q> might be preferred:

The quote element is more semantically explicit than quotation marks
The quote element can be styled (as noted by @Juhana)
The quote element allows for characters other than '"' (the guillemet  - « - for example)
The quote element can be used with the cite attribute to explicitly link the quote and the source

