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I totally understand what is happening with you. I have participated in at least 500 contests so far, and won only 8.
The Main reason is that the level of Competition is very high. Even if you got a 5 star rating you still can't be sure about winning the contest.
There is no one forcing you to participate in any contest, you choose the contest which attracts you, the contest you really want to participate in, even if it has a little low price money. ( Thats the coolest thing about being at Freelancer ), you would find 4-5 contests like that everyday, so give your best. Winning small is better than winning nothing.
Another important thing to remember is that, keep yourself calm even if you get rejected. Forget it and move to another. Do not get angry. I know its hard to control anger at such moments :P

Good Luck.

It is not the software that is important, but your design skills. No one software is better than another, it's all subjective, and you need to find what software best aids in your creative process. If you want to try some other software you can look at Sketchbook Pro or Clip Studio Paint. However these packages are not exclusively vector based like Illustrator is, but do have vector components. I personally like to draw a lot of my custom work as raster, then take it into Illustrator and vectorise it. But this is just my work flow, and not necessarily something that will work for other. Good luck!

I have to say that software is not important, the design and creativity behind is what it is important. Work with the software you're more comfortable with and the one you have more experience with. I agree as well that the most important thing is your design skills, these will be shown on the design you deliver to contests and projects.
If you're still worried about software keep practicing on Adobe Illustrator or Corel Draw, these are the top at the moment and when you get the best out of them along with your skills, clients will come to you :)

One does not simply win contests by getting familiar with a certain program, my friend. What if i told you, that awesome logos can be created using nothing, but a piece of paper and a regular ball-point pen? or if you want digital, MSPaint is good enough, too?

Like JoshuaRegan said, its your skills that will earn you fame and fortune. A software is just another tool.

You have already invested two months learning Illustrator. Now, invest a little bit more daily to reinforce what's learned. If you feel you have passed the beginner level, search the internet for higher levels of tutorials.
Continue training your hands, eyes and mind by constantly creating. Don't create just to win/earn money but create to turn the very act into your passion. Keep raising the bar gradually, and pretty soon, you'll find yourselves with lot of original artwork which you can exhibit in your portfolio. This may in turn lead to winning bids in projects, when employers check your profile.

If you still want to try out another graphic program, try CorelDraw, Photoshop or the free editor, Gimp.

All the Best!! :)

Illustrator + photoshop. And search for this tag in google :"how to use Smart objects" and after that go find mock-ups for you.

All said above is true . Illustrator is one of the tools . Every day look at other logos , highlight the best in your opinion . Paper and pencil is the best tool for creativity . Good luck !