We measured the same water with the turbidimeter and turbidity tube and received two different results. Why?

Turbidity meters are more accurate and more precise than turbidity tubes.  However, they are also much more expensive.

A turbidity meter sends a beam of light into a sample and measures the amount of light reflected by the particles.  It is considered quite precise, but is limited to the small volume of sample in the sample vial and so is sensitive to how a sample is collected and whether it is a good representation of the water you want to know about.  It is a good idea to take several samples (e.g. three) and then record the average reading.   

With a turbidity tube you are measuring the depth of water that light will pass through after it is reflected from the base of the tube.  This is less precise than with a turbidity meter, especially for low turbidities.  It is sensitive to the ambient lighting, the colour of the background you are reading it against, the colour of the water, and the person’s eyes doing the reading.  Different turbidity tubes from different manufacturers use different formulas to calculate the gradation scale (the lines on the side of the tube).  Rounding to the nearest line also brings inaccuracies. 
 
Turbidity tubes allow a rough and ready approximation of NTU, and are good for comparison between samples and over time (assuming you use the same tube and all else is equal). If, 5, 20 and 50 NTU are the most important measures you want to gauge (e.g. for chlorine or BSF use) then you can recalibrate the tube to match the meter (as in mark a new line on the tube with a sharpie).