DI
A DI box is a box that is used when you want to connect a high impedance, unbalanced output - e.g. an output jack from an instrument (acoustic guitar, electric guitar, bass, keyboard etc) to a low impedance balanced input, e.g. a mixer input, a microphone input on a preamp.
All this little unit does really is cut down on noise, distortion (and possibly ground loops) that may arise when connecting two devices of different impedance and connecting unbalanced outputs to balanced inputs.
There are 2 different types: active and passive. Active differ in that they can also amplify the signal, and therefore need power.
More expensive passive (and active) DI's typically use a nice transformer which aside from being less susceptible to hum, many audiophiles are convinced they give a 'nicer' sound.
BUT a DI will not make your acoustic sound better when plugging it into your amp.
In fact, an amp expects a high impedance, unbalanced signal (otherwise we would all need to use DI boxes before plugging in our guitars) so a DI may even be detrimental.
Acoustic Guitar
The problem you are having is one of accurately capturing the sound of the acoustic. When you play the acoustic, you hear everything that is coming out of the sound hole - loosely, the sound is shaped by all the properties of the acoustic guitar and the room you are in.
When you plug in your acoustic, you are relying on a little pickup (often a 'piezo transducer' if it is the pickup that came with the guitar - one of the cheaper varieties) which is usually attached on the inside, under the bridge, to receive the sound and send it to the amp.
Obviously, it won't sound the same and generally doesn't sound as 'nice'.
So, one step could be to upgrade the pickup that is inside the guitar. This wouldn't be cheap. Probably around £60 - £200 just for the pickup.
The second option is to look at your amp. It's a very nice tube amp, so you will probably need to fiddle with the gain controls so you dont introduce too much distortion. Turning up the 'pre' gain control should make the signal higher before it hits the tube, driving the tubes harder and hence more distortion..
You can try fiddling with the EQ. Often acoustic guitars sound a bit too honky so you might want to experiment with the 'mids'. If you can select the frequency of the EQ, then try turning the mid right up and sweeping through the frequencies until it sounds awful. Then you have found the frequency you want to cut...
If the EQ is more basic than that, you might want to look at getting a more comprehensive EQ pedal.
But try fiddling around with EQ...it's fairly easy to get going, you just need to be thorough. Locate the frequencies that do not sound nice and cut them until the overall sound improves.
If you can describe what is 'not nice' about the sound, or post a clip, perhaps we can find some good EQ starting points.
Also, fiddle around with that spring reverb. One thing a pickup'd guitar misses that your ear hears is the 'sound' of the room. Fiddle around with reverb to make it more sympathetic and 'natural'
You can also look at acoustic guitar amps. I've only played with a few in a shop and TBH, I didn't think they offered a much more 'natural' sound than a regular amp...but I dont know much about it, so look around
BTW,
I'm saying all this assuming you are plugging in your guitar because you are playing live.
If you are recording, use a microphone. The absolute most natural results you can get is by using a condenser (or ribbon) microphone (or 2) to record your guitar.
