Turn logo's white background to alpha transparent and clean up fuzzy edges

I have also tried turning it into a vector image using Illustrator, but the straight lines become distorted:

You can turn a logo shape into a vector shape fairly quickly in Photoshop

Use quick selection to select the logo

In the paths panel click on  “ Make work path from selection”

Go  quickly round the path and adjust any points that are missing the shape at all. On straight edges the easiest way is to delete any unnecessary points.

In Pen tool options : Select Make : Shape

Select the shape tool and adjust the stroke and fill colours as required

Almost instant vector
 
 
 
Hope that helps

Dave

The layer blending options slider with these settings worked fairly well. (Note that the highlight slider is split.)

I added an empty layer underneath and merged the two layers to make the transparency permanent.

Filter > Noise > Reduce Noise. Increase strength and check JPEG Artifacts.

Then touched up curved and diagonal edges with the Blur tool at a low setting.

That said, with a logo like this that would be fairly easy to trace with the Pen tool. That would give you a nice vector logo with perfectly smooth edges and solid fills.

It's easy to make a mask from channels here: Duplicate red channel. Then copy green channel (select all - copy) and paste it to red copy, then fade -> multiply. Make same with blue channel. Then invert your channel and adjust levels so that you get pure black and white, artifacts are gone. Use this channel to make a mask for a logo layer. Then you can make colors even in edges of logo just painting or filling... I'm not sure if you need all channels with this particular logo, but I told this if there will be some others in the future for you.

But then you only have logo file this small. This logo is very simple to draw in Illustrator and does´t take more than couple of minutes, so I'd spend those minutes with it.

Option 2 should work, but you would be better off using an uncompressed logo in the first place. The mosquito artefact aren't too bad in the grey areas but look bad on the red. Use Topaz Dejpeg to disguise the artefacts or if you don't have that use the magic wand tool and select all the red letters and fill them with that shade of red.  Then use blend if to remove the background followed by a small amount og Gaussian Blur to smooth. It won't be perfect as the compression damaged the original unfortunately.