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What is the difference between the words attrezzi, utensili, arnesi, and strumenti? My understanding is that all of these refer to tools, with the latter being used to also describe software tools. Furthermore, what other words can be used in Italian to describe tools?

I think it's useful to look at Treccani's definition of each term to find out the subtle differencies (if there are any).

Starting with attrezzo, Treccani says:

attrézzo s. m. [dal fr. attraits, part. pass. sostantivato plur. di attraire, che è il lat. attrahĕre «attrarre»]. – 1. Arnese o
strumento qualsiasi: un a. di cucina, di bottega; per lo più al plur., complesso di arnesi occorrenti a un determinato uso: gli a. del falegname; a. rurali; a. teatrali; a. navali, il necessario per l’armamento di una nave. In partic., nell’industria, oggetto di forma semplice, che serve a compiere una data operazione in un ciclo lavorativo (meno complesso di utensile).

Here attrezzo has a generic meaning and can be also translated as something used for a particular use or hardware in the industry field, for something that is needed to perform a certain operation during a working period ( less complex than an "utensile").

For utensile:

utensìle (meno com. utènsile) s. m. [uso sostantivato dell’agg. prec.: già in latino esisteva il plur. neutro sostantivato utensilia
«cose utili spec. per la vita domestica, utensili» e su esso è fondata
l’accentazione piana utensìle]. – 1. Nome generico dei varî
arnesi che servono agli usi della vita domestica, spec. in cucina: u. domestici; gli u. più necessarî in cucina sono le pentole,
i tegami e le posate. In botteghe artigiane, laboratorî, officine,
ogni attrezzo semplice, atto alla lavorazione dei metalli, del legno o
della pietra; per es. il martello, lo scalpello, le tenaglie, le
pinze, le forbici, la sega, la lima, il trapano, il punteruolo, ecc.

An utensile (utensil) has a more specific meaning especially referring to home life for everything you use in the kitchen, but also, for example in a craft store, the hammer, a pair of scissors, the saw, the driller, etc.

For arnese:

arnése s. m. [dal fr. ant. herneis, harneis, harnais «provviste di viaggio, vettovaglie», poi «armatura», voce di origine germ. (ant.
scand. *hernest)].
3. Nell’uso com., nome generico di masserizie, oggetti varî, utensili, strumenti, ecc.: molte robe su per le stanghe, secondo il costume di
là, e altri assai belli e ricchi a. vide (Boccaccio); i sacri arnesi
Che prima ritrovâr Cerere e Pale (Parini); gli a. del fabbro, del
falegname, ecc.; aveva con sé tutti gli a. del mestiere. Anche,
qualsiasi oggetto che non si voglia determinare o di cui non sovvenga
lì per lì il nome: che fai con questo a.?; dammi quell’arnese.

Arnese is even more generic, indicating various types of goods and objects

For strumento:

struménto (letter. istruménto; ant. instruménto, stroménto, storménto) s. m. [lat. instrumĕntum, der. di instruĕre «costruire,
apprestare»]. – 1. Genericam., arnese, congegno, dispositivo e sim.,
necessario per compiere una determinata operazione o svolgere una
attività

Finally, for strumento it is something that is needed to perform an operation or an activity.

Please note that in every definition the other terms you mentioned are used as synonyms, in order to explain the concept. In fact, looking at Treccani synonyms dictionary you can find:

attrezzo≈ arnese, strumento, utensile.
utensile ≈ arnese, attrezzo, strumento.
arnese ≈ attrezzo, ferro, strumento, utensile.
strumento ≈ arnese, utensile.

As correctly pointed out by @CasaMich another word user for tools is ferri:

In partic.: a. I f. del mestiere (o di bottega, o assol. i f.), gli
arnesi usati nei lavori artigianali, come martello, lima, cacciavite,
tenaglie, ecc.; fig., quanto serve, anche se fatto d’altra materia,
per un dato lavoro o genere di lavoro. Analogam., i f. del chirurgo o
chirurgici, gli strumenti necessarî per gli interventi chirurgici;
anche assol., nelle espressioni andare, essere, morire sotto i f., e
sim.

in expressions as ferri del mestiere (tools of the trade) o ferri del chirurgo (surgeon's tools).

This is a really hard question with no simple answer, mainly because of the degree of heterogeneity of word usage that one can find in spoken Italian among different regions. I myself am biased, coming from Rome. This is surely not going to be an exhaustive answer, although I hope it can give you some clues.

As you have correctly inferred, they are used in a somewhat interchangeable manner. This can clearly be seen by a standard dictionary definition. For example, looking up the word "attrezzo" in Treccani yields

1. Arnese o strumento qualsiasi.

That being said, certain contexts have with time adapted more the the use of one of the terms with respect to the others. Also, the terms might be interpreted as referring to different levels of concreteness i.e. do you want to stress the general concept of a tool or are you point out at the physical object itself? How complex is the object? The answers to these questions somehow gives us hints on what a good usage could be.

Attrezzo

Almost every Italian household (at least in the Center-South) will have a (and will refer to it as) "cassetta degli attrezzi", meaning a toolbox where one keeps screwdrivers, hammers etc... In this context, these tools are referred to as "attrezzi", but could easily be referred to as "arnesi".

Utensile

In common usage, I would dare say that this word is very commonly used in a kitchen context. When referring to pots, pans, cutlery ecc... you would almost always use the term "utensili da cucina". Also, in a workshop context (such as an artisan's shop) you would refer to the tools used as "utensili" (although you could without any problem use the words "strumenti" or "attrezzi").

Arnese

This is the one I find most interesting. This is a word that evokes generally variety, generality, and somewhat opacity when it comes to listing the exact tools and their usage.

Prendimi quell'arnese/attrezzo là!

This expression could be used when I'm asking a friend (not of the trade) who just came in my mechanic shop to pass me an object who's name is unknown to my friend. It somehow has this blurred connotation to it, either because I'm too lazy to name the object or because I know my friend would not know what to pass me if I said the exact name of the object. Attrezzo sounds more specific.

This is especially notable in certain regional expressions. In the area around Rome, you could hear the following phrase yelled at a computer

Ma come funziona (que)'sto attrezzo?!

to show unfamiliarity with the object and almost disdain.

Strumento

This is kind of an all-encompassing term which can be used in most circumstances.

Gli strumenti del mestiere

means "the tools of the trade", although not referring so much to the skills required for a job as the actual physical tools.

This terms can be also be used in an extremely generic way though:

Mi rifiuto di essere un tuo strumento!

As a general idea, the word "strumento" is used when one wants to emphasize the fact that the tool is being used for some purpose.

You are also right when it comes to the used of "strumenti" in a software context, as this is the translation you will find in most available programs.