You are absolutely right - running and running fast are two VERY different things. Luckily, there are several tips listed below that you can try to help increase your speed!
First off, focus on your current training schedule and ask yourself how often you are running, and how far. To get faster running times, you first have to build up your base. This means that your body needs to be used to working hard for extended periods of time. You need to be in good cardiovascular shape because running faster equates to an increased heart rate, and can be very difficult to sustain if your body is not used to it. Start incorporating other types of runs into your weekly training, such as tempo runs, hill repeats, and long, slow distance runs. A variety of training runs translates to improved cardiovascular shape, and you will be better primed to start picking up your pace.
Next, focus on your form. A lot of runners' paces and race times suffer simply because of poor form. Poor form means that you are wasting precious energy on how you are running when, if you cleaned up your form, your energy could be more focused on increasing speed. Good running form looks like maintaining a short quick stride (avoid over lengthening your stride) with arms bent at a 90 degree angle, neck at a relaxed, comfortable position focused straight ahead, and hands relaxed but core kept tight.
Next, take to the treadmill. Sure- at times, the "dreadmill" just totally sucks. But it is perfect for helping increase times because you can actually set it to the exact pace you are trying to hit. In this way, your legs and body can get used to what running at that pace feels like. It will help you gauge how quick your leg turnover needs to be, and how much work you may or may not have left to do in order to run at that pace during a race or out on the roads and trails.