I think my neighbour stole my cat!! HELP!!!

Did you make up any "Missing/Loss Cat" flyers with a photo when your cat went missing?  If so, you could take a flyer to your neighbor and tell them that you are going door-to-door asking if anyone has seen your cat.  If they have your cat and not just a look alike cat, that would very likely cause them to give the cat back to you.

Your neighbor didn't necessarily steal your cat.  Many people see a cat roaming around outside and start caring for it, mistakingly thinking it's a stray.   At any rate, even if you don't really believe that,  it does give you a non-confrontational way to approach your neighbor.  "Hi, your cat looks so much like my cat who took off about 3 weeks ago and I'm wondering if you may have been taking care of her all this time?"   This will hopefully provide you with an opportunity to see the cat up close so that not only can you check to be sure it's your cat, the cat itself might provide proof by greeting you with obvious affection and recognition. 

If the neighbor honestly believed they were caring for a stray cat,  this may be all it takes to get your cat back.

Things become much more tricky if your neighbor insists the cat is theirs.  Do you have any photos of your cat that you could use for identification purposes?  That might help you with any official complaint, should you decide to file one.  The problem, as you point out yourself, is that your kitty isn't microchipped so it ultimately becomes your word against that of your neighbor.  

I'm sorry you're going through this.  I hope things get worked out.  Ideally, I hope your kitty shows back up at your door soon, having just had an adventure and now being ready to settle down. 

Is your cat trained to her name? If you call it will she come?

Do you have photos or any kind of paperwork? Vet bills, adoption certificate etc. These can all act as evidence.