I ask this question because I have been on a low-calorie diet and have been working out on my total gym once or twice a day and walking 2-2.5 miles a day five days a week for the past two weeks. In the mirror, I can clearly see muscle definition and fat loss, but my weight has only decreased by but a few pounds and, some days, it has actually gone up a pound. I am aware that muscle is heavier than fat and that muscle has a tendency to retain water. I also made it a point to say "all things being equal" because I am aware that how fast muscle builds vs. how fast fat burns depends on the intensity of your workout and also your diet. However, if my muscle tissue is building faster than I am burning fat, that would introduce an additional factor that could explain the erratic numbers I see while I am on the scale.
Muscle does not build faster than fat burns.
It sounds like your main issue is not that you are not burning fat or gaining muscle, but instead you're not actually measuring those things. Your body is not entirely muscle OR fat. In turn, when you weigh yourself you are not only measuring those two factors. You're measuring fat mass and lean body mass (muscle, bones, the food you ate for lunch, etc).
Simply, if you changed your diet and increased exercise you will most likely improve your body composition. You said you see that in the mirror; I would trust that over the scale. You can also use your physical fitness as a good measure -- are you lifting more, running longer, etc.
If you really want to continue to use the scale, then I agree with Kneel-Before-ZOD. Do it weekly or biweekly on the same day at the same time of day. And don't stress about it.
Bodybuilders tend to interleave periods of 'ripping' with periods of 'bulking'. If you want to gain muscle fast, you will need to eat big time and will have to accept also 'gaining' a bit of fat and you will need to do mostly anaerobic exercises and skip cardio. If you want to loose fat quickly, you will have to undereat a bit and work out with lower weights higher repetitions, etc. You will need to accept loosing a bit of muscle. And yes, if you focus on the larger muscle groups, have a good bulking diet and work out schedule, you could gain muscle faster than you could loose fat when on a diet. Going on a bulk/rip cycle can be an amazing way to get in shape. Start off with bulking until your body fat percentage stops going down. Once you are at that point, start interleaving periods of ripping with periods of bulking.
Ok let's do some math.
One pound of fat is about 3500 kcal, and is about 45 cl or 15 fluid ounces in volume. You don't mention your weight or your diet, but let's say you've lost 1000 kcal/day, which is a very high rate of calorie loss, in two weeks, that's 4 pounds, or 60 fl. oz./180 cl. This might sound like a lot, but considering the fact that your burn fat from all of your body, it's not a lot of fat.
On the other hand, whenever you eat less, you will lose a lot weight in solids and fluids, this will also fluctuate a lot, you can assume that the error margin at your weigh ins are somewhere around 2% of your body weight, so having days of increased weight is perfectly normal.
Regarding muscle building, as others have said, it's a lot harder to put on a pounds of muscle than it is to lose a pound of fat. If you stay the same weight over a year, 10 pounds of added muscle weight is a very good result if you're completely new to lifting, if you're a man, and if you rest and lift well. That's less than a pounds per month! If you lose weight, it's even harder.
I have been working out regularily for over a year. I initially saw lots of fat melting away, but my weight went up by 4lbs 2k aprox in the 1st three months. In the end, by measuring my fat (pinch test) before and after. I gained equal to 7lbs in fat and gained 11 in muscle. I did not change my diet in that period. So in 3 months I lost 7lbs (5% for me) in fat while gaining 11lbs overall, everything stayed stable after the first 3 months.
As a woman, it almost made me really angry to see that I was gaining so much while working my butt off. But I could see the fat melting off quickly, so I didn't panic.