Your neighbor that is losing a lot of weight without working out is (I am guessing) NOT someone whose advice you should take. Only because in my observational experience, women that lose a lot of weight like that either A) have those Gwenyth Paltrow metabolisms, B) are not eating ANYTHING or C) perhaps younger where their metabolism hasn't changed and they can just burn off food.....in my mind I want to hurt people like that LOL.
Also as usual, everyone's body is different so one workout combination might work intensely well for some but horribly for others. I know this very well from my own experience. Forgive the length of this post but here is what I went through and I'm so glad to be where I am now ~
I originally lost about 10-15lbs like 10-11 years back when I was 26. I'd moved to FL and partied a lot and gained - was up to 176!!! Back then was the Susan Powter book so I read that and everything was about fat-free etc. and I started cooking that way and working out for the first time in my life. I got down to about 155-158 and everyone was like, whoa you look great. I kind of parked there for a while. By no means was I fit, just not looking hefty anymore.
I graduated college (a bit delayed obviously!) and moved to Chicago, got a job, started working in offices and eating more and moving less. Clothes got tighter. Etc. After a while I started realizing I needed to lose again. Started eating Lean Cuisines and doing Taebo. Then I discovered Body For Life; threw myself into that and was doing weights and Taebo.
But you know what, for several years I was working out fairly regularly and eating healthy foods but I saw NO muscle definition that I wanted and no weight loss. I was stymied. And frustrated.
It wasn't until about 3 years back when I FINALLY started journaling what I ate that I realized that was my main impediment to seeing results. I was eating healthy stuff but way too much of it. So once I started watching, I started losing.
I then was only doing Taebo. I added the FIRM workouts. Started to see tone and weight loss; slow and steady. This year I added Cathe and now Yoga/Pilates.
**I am writing all this down not to bore you but to show you that I had to experiment what worked for me. Earlier this year I tried really "Heavying up" my weights~ works tremendously well for many women. For me, I was bulking up big time in my lower body. So I evaluated my program and went 'back' to what I was doing and adjusted to lower weights for LB and still heavy for UB. I'm currently doing a lot more cardio to encourage the last however many pounds I want to lose.
I watch what I eat every day. I have 1-2 cheat meals a week but it is working out regularly and consistently that I believe allows me to maintain what I have achieved and not turn into a flabalanche when I stray and pig out at a cookout!!
I would suggest you definitely NOT lose the weights. Lifting weights has so many long term benefits plus it can chisel your body to look REALLY NICE. I cannot tell you how much I enjoy wearing tank tops and such because I'm proud of my arms; finally!!
I hope that my experience will help you to realize that other peoples opinions might be helpful but do what works for YOUR body; and that will take some experimentation. However I would second what RedFlame says and tell you to definitely keep the weights whatever you do!!