BPM and tempo are essentially the same thing, except that BPM is more specific. Tempo just refers to how fast a song is played in general, whereas BPM is the specific unit of measure -- beats per minute. In a car, it's kind of like the difference between "speed" and MPH (or Km/h).
When you are first learning a song and it may be difficult to play, the true tempo of the original song may seem fast because you're still concentrating on learning the song. It's is often a good idea to learn the song at a slower pace and then gradually speed it up to the tempo of the original recording (from the original artist) as you learn the song. Often what happens after that is that, once you really know the song by memory, you may have a tendency to play the song too fast -- unless you remember the original tempo of the song and set the BPM to the correct value for the style on your keyboard. And then, when you really know the song, playing it at the actual original tempo may feel slow!
Of course, another possibility is the accuracy of the information you are looking up. When you say that you look up the tempo/BPM of a song, where are you getting that information? If you are not sure, you can just play a recording of the original song, get a stopwatch or at least a watch with a second hand, and then count how many beats (usually quarter notes, but it depends on the time signature of the song) are in a specific amount of time, and then calculate the BPM. For example, you can count the beats in 15 seconds, and then multiply that value by 4, since 15 seconds is 1/4 of a minute. Or count the beats in 20 seconds and multiply by 3. Or count the beats in 30 seconds and multiply by 2. Or count the beats in 60 seconds, and then that is your BPM directly.