Well, like in English there are different ways to say <em>to use</em> (<em>to utilize</em>, <em>to make use of</em>, <em>to employ</em>, <em>to apply</em> etc.), there are also several ways to express the same thing in German. In general all words mean to use something (abstract or concrete object) in order to achieve something. But when applying them there are some differences.

You use <em>benutzen</em> when using concrete objects (<em>Zahnseide benutzen</em>, <em>Klopapier benutzen</em>) but it's also possible when talking about abstract things (<em>Benutz deinen Verstand</em>). I guess this word is most colloquial. In some regions the word is written and pronounced with an <em>ü</em> instead, i.e. <em>benützen</em>. You can often just use <em>nutzen</em> and <em>benutzen</em> interchangeable (<em>Ich nutze[=benutze] das Auto</em>) but, for instance, when talking about using a service or accepting an offer you can only go with <em>nutzen</em>:

Wir sollten das Angebot nutzen. So, <em>benutzen</em> is more in sense of <em>to utilize an object</em> and while <em>nutzen</em> can connote this meaning as well, it often means to take an opportunity. (<em>Das schöne Wetter (aus)nutzen)</em>

<em>Verwenden</em> is often interchangeable with <em>benutzen</em>. It appears to be somewhat more formal.

Zum Reinigen der Zähne verwende[=benutze] ich Zahnseide.

Beim Kochen verwende[=benutze] ich ausschließlich asiatische Kräuter.

Though technically possible, in some contexts it may sound odd to use <em>verwenden</em> instead of <em>benutzen</em>:

Benutz[Verwende] deinen Verstand.

<em>Anwenden</em> means to make use of something (abstract or concrete) in order to achieve something. <em>Einen Trick anwenden</em>, <em>Gewalt anwenden</em>.Bisher bin ich noch nicht in die Situation gekommen, dass ich Pfefferspray anwenden[=gebrauchen] musste. Note, here again <em>gebrauchen</em> means to be useful while <em>anwenden</em> means to make use of it.

As you see, there's a huge overlap in sense and you have to learn the tiny differences by speaking German.

I hope that the info helps you!