I gather from the aspect ratios (top one is 3:2, bottom one is 4:3), that the top image is the dSLR one, and the bottom image is the one from your TZ40. And at web sizes, while there's some improvement in image quality with the dSLR, it's not a huge amount better, and some could be compensated for with post-processing, rather than using straight-from-the-camera JPEGs. While your camera doesn't do RAW, that doesn't mean you can't make adjustments in post.

The main problems you have are slight overexposure (probably from the sky), and the fact that you're shooting a high dynamic range scene--that is, a scene where the range of values from white to black is larger than your camera's sensor can record. While a larger sensor can capture a larger dynamic range, it, too, will have difficulty with this kind of scene: notice how in both shots, detail has been "blown out" (overexposed) in the clouds. Having a dSLR would not necessarily have helped you here as much as technique.

Things you could have tried:

1. Your camera has an iHDR mode as well as a High Dynamic processing setting. You could use those, or you could have bracketed exposures and then combined them in an HDR or exposure fusion package in post to cover the dynamic range your camera's sensor cannot. A dSLR user would probably have to do this as well.

2. Your camera has PSAM modes and a histogram. A histogram charts the number of times a value from black (left) to white (right) occurs in the image. This will let you judge the exposure another way, and you can see if you've blown highlights or lost shadow detail and adjust at the time you're shooting. The PSAM modes give you the same control over exposure that a dSLR typically can. This is the histogram from your image:
   
   
   
   That peak at the right side, and the flatness on the left together indicate overexposure. You had some wiggle room to bring the exposure down. Your friend's histogram looks like this:
   
   
   
   He used a lower exposure, and filled in more of the range, but still had overexposure in the clouds, indicated by that peak on the right. When values get jammed up at either end (or both) of the histogram, this indicates you're looking at a high-dynamic range scene. You may have to make a choice between the highlights or the shadows, or shoot bracketed shots and merge them in post.

3. Your camera can do post-processing in-camera. While not ideal, and yielding far less control that doing post-processing in a dedicated computer application like Photoshop or the Gimp, you can still mess about with how the image is going to look in-camera. Even something as simple as boosting saturation (Your camera has a "Happy" Color mode that does this) can make a large difference in how the final image looks.

But probably learning to post-process is your best bet, and here, you may eventually want to move to a camera with RAW capability. However, I don't think a new camera is called for yet. Just more practice, knowledge, and gaining some post processing skills.