Would a DI box improve my guitar sound?

Have no experience with them. Would a good pro-quality DI box reduce buzzing and improve the general tone of my Telecaster into my Babyface?

A DI, as noted above, is mainly for impedance matching. If you have an interface with an instrument input where the impedance of that input is too low, then you will lose high end. If you have any FX pedal with buffered bypass (like Boss pedals and many others) you can see if you hear any improvement with the pedal in the middle.

a real amp,
would improve the sound of your guitar into your babyface.
 
your mileage will not vary.

A DI is a device mainly used to convert a guitars hi impediance out to a lo Z to send a clean signal to a mixing board. There will also be a hi Z output that goes to your amp on stage. In a studio one might use one in conjuction with a miked up amp for the engineer to have the clean sound. 
A real good DI like one made by Radial or Jensen might make anl improvment to the signal because they use better transformers ( Jensen) , A cheapo DI will make no difference comparied to your interfaces Hi Z. Might make it worse. 
There are many guitar pre amps that work without a speaker. My Blackstar as an example. 
The best tone is going to come from a real guitar pre amp. There is a sort of reaction between the PU's and the pre amp. I find plugging in my G&L directly to my interface is lifeless. 
There's certain types of guitar sounds that you can get away with that but not the type of music I play..I need the amp or the pre amp. 
 

In general, my experience between an interface HIZ and a DI box has yielded no real difference.  A DI will give you the ability to lift the ground - which may or may not do anything to help your situation.
 
 

The Babyface's Hi-Z instrument input is 470KOhms according to the specs in the manual. Which is perhaps a little on the low side, 1MOhm or higher being generally considered a better match for the input impedance of a valve amp.

Having said that, 470KOhm is also the input buffer's impedance of a Tube Screamer and people generally don't complain about Tube Screamers affecting their tone for the worse when switched off. If you've any pedals that have buffers between guitar and interface the guitar will see that buffer in any case, not the Babyface's impedance. A good DI box with a 1MOhm+ input impedance might get you a bit more volume and treble, but so would any Boss pedal put between guitar and interface (Boss pedals all have 1MOhm buffers which are pretty good as it happens).

The additional buzzing compared to what you might get in stage is almost certainly down to being near a computer. Computers can emit all kinds of radio interference which single coil pickups are very susceptible to receiving. Add flourescent/low energy lighting, domestic lighting dimmers, switching power supplies, electrically noisy transformers in monitor amplifiers etc. into the equation and noise can quickly get out of hand.

A DI box won't reduce noise unless there's a ground-loop between two mains-powered pieces of kit when the earth-lift switch on the DI might make a difference.

The best answer to single coil noise is to first try having the guitar at different angles to the computer, monitor screen etc. and see if that helps, or to get the guitar further away from the sources of electrical noise. It might also be worth shielding the pickup and control cavities with copper foil or conductive paint connected to earth, though Fender have been applying conductive paint to the cavities in many of their instruments, even the less expensive Mexican ones, for some time. The usual give-away that this has been done at the factory is a wire screwed into the pickup or control cavity, or trem spring cavity on a Strat.

As for Teles and high gain, a Tele+fuzz->Marshall setup is as old as fuzzes and Marshalls. Or overdriven tweed Fender amps. You'll probably always get some noise, because a Tele bridge pickup in particular is quite a bit more powerful than the anaemic Strat vintage single coils, and a Tele Texas Special is about as powerful as a PAF humbucker and can really buzz away.

It's even quite possible the guitar is just as noisy on stage - it's just that playing live we and the audience tend to filter out the noise, it's only when recording or listening to a recording we spot it.

sharke
Have no experience with them. Would a good pro-quality DI box reduce buzzing and improve the general tone of my Telecaster into my Babyface?

A "vanilla" DI box won't really affect the tone.
A-Designs Reddi Box and Avalon U5 are examples of DI boxes that are designed to enhance the sound.
 
When recording near the computer, true single-coil pickups are going to pickup EM noise.
You can minimize it (shielding the pickup cavity, turn away from the computer when recording, etc)... but you can't completely eliminate it.
 
Rather than a higher-end DI, I'd checkout some "noise-less" pickup options.
This won't completely eliminate noise... but it'll stop most of it at the source.
 
MusicMan is using an active noise-reduction circuit in their single-coil guitars.  
Works surprisingly well...

For my Strat, changing to some Fender Custom Shop noiseless signal coils totaly worked with sitting close to my workstation. It did not ruin the Strat tone and now it's almost dead quiet.  Many companies make noiseless signal coils. 
My G& L Astat ( Tele)  which had SC soap bars was also very noisy. I put some GFI humbuckers but it changed the tone to much so I'm looking into something better. Harder to find Soap Bar's. But there are many regular Tele upgrade Pups on the market. If I change the pickguard I can just use regular Tele Pup's. 

I have EMGs in an AM BC Rich ST in a SSH configuration. They are OK, very quite. I have Bill Lawrence PUs to replace those, but haven't installed them yet. I put Joe Barden PUs in my strat and like those. A friend has Lace Sensors, but they have too much bite for my taste. As tlw says "personal preference".

a DI has questionable benefit in your case..
you can..
a) copper foil shield your cavities
b) change to noiseless pups
c) get a pod, modeling setup, preamp or other means to quietly record..
 
but, at the end of the day, it is far easier and cheaper to use your phone as a remote for Sonar and play in a position and orientation in your studio that mitigates noise..
 

I agree with pretty much all of the above.  If your preamp accommodates both mic and instrument level signals it probably won't make a difference.  I say probably, because since you are playing a tele we are all thinking single coil pickups and that's usually where the noise sneaks in.  Do you get the  buzzing when playing through both the neck and pridge pickup (center position)?  If so, I would take a moment and diagnose that first.  Try a guitar with a humbucker and see what you get.
 
Also, if you play out at all, it can come in handy to have a DI in your gig bag for those occasions that it's needed.
There are plenty of quality inexpensive ones available these days.  I just picked up a Radial Stage Bug that is phantom powered for times that I use a Piezo pickup.
Also, a DI can get rid of hum from a ground loop, if that is an issue.

Sharke,
 
This was recorded with a Neve RNDI, the new one for $269.  Not sure whether it's $269 of goodness, but it does warm it up a little.  The problem with these things is that it's so subjective.  Listen to the solo at the end, it gives a better idea than at the beginning.
 
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