Counting Beats:
This is the number of beats in the measure and it could be the different ways to count the time in a measure.
In 4/4 time; 1 2 3 4 (or could be counted like this) 1 & 2 & 3 & 4 & (or like this) 1 e & a 2 e & a 3 e & a 4 e & a …

Counting Note Value:
This has more to do with whole notes, half notes, quarter notes. The note value of a whole note would be 4 counts (in 4/4 time). The note value of a quarter note is 1 count (in 4/4 time).

I hope that makes sense to you. maybe someone else can explain it a little better

If the time signature is

4
4

then the quarter note get the beat

The easiest way to explain is by clapping. If you are clapping and counting the beats in a melody, you would count 1 2 3 4 the whole time you are clapping, but your hand would stop clapping at the start of each new beat

For example, if you have a whole note, then you would clap one time, but count four beats