Tone is something that typically takes a while to develop an ear for, and extracting specific tonal qualities from sounds is a skills set in its own right. Tone can add an extra layer of interest to your production and it's worth paying attention to. There are some producers that are much more conscious of tone than others, the producer that comes to mind when I think of investing time into getting specific sonic qualities out of instruments and getting "that sound" is Pretty Lights from Boulder, Colorado. He invests in a lot of analog gear to get a fat, warm sound from every instrument in his mix, see the picture of part of his studio setup below. You really hear this in his music, which is very distinctive as a result.
Now obviously you don't need to go to the extreme of spending thousands of dollars on analogue
equipment, but being conscious of the tone and sonic qualities of the instruments in your mix is definitely a good thing. Having some outboard gear never hurts, you could invest in an analogue synthesiser for making fat, warm and well rounded subs. Or you could invest in a character compressor. Many recognised producers have a hybrid analog / digital setup with outboard gear to add character amongst other things, and one of the most common things to own has to be a compressor, limiter or both. By definitiion, this will have a different sound to it's software equivalent, we all know this. This is due to things like the lookahead function in software as well as the fact that we are talking about voltage in analogue compared to code in a digital setting. For a relevant example of this, check out Teebee's studio, picture below. Two limiters and a compressor are amongst the outboard gear he uses. In no particular order, these are a Solid State Logic Gbus compressor, an A Designs "Nail" Limiter and a Pendulum Audio Limiter.
Other things you could look at intergrating into your setup (and popular in Drum and Bass and Hip Hop) are old samplers like the Akai and EMU range, including things like the old MPC60, a often sought after sound. In the picture above you can see on the left hand side rack an old 90s era EMU E6400 model sampler. Units like these will add subtle analogue compression and give a bit of a crunchy, low-fi digital feel to sounds, depending on the make and model.
Even finding an old cassette tape deck and recording sounds to casette can help get some character into your sounds, you don't need to have a lot of money to do this. Of course you can always use character software, tube and tape saturation emulation plugins come to mind. FabFilter Saturn, the Waves Bundles and if you can afford them, UAD plugins are all good examples of this but there are hundreds... including much freeware that sounds very disticntive and unique if you know where to look. Even something as simple as an EQ can be used to add a particular sonic quality to sounds. To do this, create a bell curve EQ and boost by five or ten decibels (watch your levels) in a range of the frequency spectrum that sounds interesting in the context of the whole sound. You've changed the tone of this sound just through this action, and this can help to create a unique take on sounds that have been used often. For example it's reasonably well known that Paradox / Alaska has done this a few times with the Amen break to create unique versions of it.
Taking this a step further, if you start splitting things into multiple bands and using different character effect software on the various bands (such as distortion, saturation or tape emulation software for example) you can get some really uniqe tones from your sounds. Furthermore, if you were to run a notch filter up and down a sound with automation, then pass that sound through some character effect plugins you'd also get some really cool results. Really, there are as many ways to refine the tone of sounds as there are producers out there. So be conscious of tone, be conscious of it on a macro level when you're mixing a track, but also be conscious of it on a micro level when designing or working on individual sounds.
One final note with regards to tone. If you are conscious of tone you will find yourself developing a taste for certain sonic qualities that you routinely extract from sounds you are using or make. Just be a bit wary of using the same processing and the same EQ choices on all the sounds in your mix, if everything has the same tonal / sonic characteristics, your job as an engineer will be made more difficult because none of the sounds will stand out from each other, there's no contrast. Often tone is referred to in terms of colour, and with good reason, if a painter used the same colour for every element in a painting it would be pretty one dimensional.