I am thinking of using a diffuser on my canon speedlight flash to soften harsh shadows when shooting indoors. I was wondering what the disadvantages of this might be.
It will reduce the total amount of light that your flash can put out. It will also use more power for the same amount of light hitting your subject.
When indoors, if your flash can bounce/swivel but the wall is further away, you will probably be better off bouncing without a diffuser.
Harsh shadows are due to a small apparent light source (light travels in a line, soft shadows are when the transition from light to dark is gradual and the object can see part of the light source).
Adding a diffuser barely (e.g. stofen) or doesn't at all add to the size of the light source. The exception is that when walls and ceilings are close, the light is diffused and sent to those surfaces which can create a bit of a larger light source.
Note that when you bounce a flash, it can give a color cast based on the color of the surface.
There's only one disadvantage that it may not have the effect you are looking for.
A diffuser spreads the light in more directions than the flash illuminates without the diffuser. Reasons why this might be unhelpful include:
People often assume that a diffuser softens the light in an image. This is not really true. Or at least, it's a secondary effect. The softness of the flash lighting in a photograph is determined by the apparent size of the light source as seen from the position of the subject. Diffusers soften light (when they do it at all) mainly by causing the light to be partly reflected from the flash off other nearby objects before hitting the subject, thus increasing the area from which illumination comes.
An alternative technique is to angle your flash head up or back, and put a card or sleeve (a "gobo") on the flash to prevent all direct illumination from the flash hitting the subject. Thus the light source is now the reflection from the ceiling (which is large) rather than the flash itself (which is small) and so the lighting is softer.
The best book I've read on this subject is On-Camera Flash Techniques for Wedding and Portrait Photography by Neil van Niekerk.
I'm not sure you can call it a disadvantage as such, but they will usually reduce the overall amount of light, so you have to consider that when metering your subject. The light loss will vary, so there's no hard and fast answer to what you may need to compensate. Anyways, a good source for information on diffusers can be found on this site, including tips on how to use them to best effect.
Anyways, in a nutshell, there isn't really any disadvantages if they're used for the purpose they're intended for.
It depends on the nature of the diffuser, but most of them will eat into your light power output one way or another, and can scatter light in ways you don't desire.
If by "diffuser", you mean one of those small tupperware-like devices, like a Sto-fen Omni-bounce, the main issue is that the diffusion of the light is not really done by the small plastic device, but by the fact that it throws light in all directions, bare bulb-style, and that that light gets bounced off a variety of surfaces. If you're not near any bounce surfaces, you've pretty much thrown a lot of light away and not really softened your light/shadows by much, because the light coming directly from the flash through the tupperware is still a small enough light source to be relatively hard. This type of diffuser can be very useful, though, if you'd prefer your speedlight to behave more like a bare bulb light (say, inside a softbox).
If by "diffuser", you mean a large bounce card, or other variety of on-flash reflector like, say a Demb Flip-It, the main disadvantage is that you can't really choose the direction of the light--by its very nature the light is still coming from an on-axis source, and the light is still coming from the camera direction. You also look kinda conspicuous. But at least you don't have to rely on bounce surfaces being nearby. [BTW, that small white pull out panel that's built into most flashes don't work well as bounce card/reflectors/diffusers because they're too small. Those are mainly to create small square catchlights in your subject's eyes.]
If by "diffuser", you mean an umbrella or softbox with your light on a stand and remotely triggered, the main issue is going to be how much of a timesink it is to set up, and how much more gear you have to bring, and how much spill control each modifer allows you. But, you can then control the light placement and get much softer light than you can with on-camera diffusers.
Typically, vs. a small plastic add-on to your speedlight, bouncing gives you more control and produce better results, particularly if you learn to flag off any direct light from the flash, but this becomes problematic if there are no nearby bounce surfaces. At that point, the little on-camera diffusers can become more useful.