So earth ground and arduino's ground same thing, isnt it
No not necessarily, it depends on how the arduino is powered. If it is through USB then it depends on how the host is being powered. A lot of systems do not have a mains ground point on them.
And just use the rx and tx pins from usb. without any other power connection from usb.
That is not going to work because the data pins also need a common ground reference point.
One explanation of the differences is if the signal is noisy or is floating. The meter will tend to average where as the arduino will read a sample anywhere in the cycle.
In addition to the measurements you have taken, what do you get with the - and arduino ground connected together on one lead and the + wire on the other. It looks like the voltage from the plasma is not isolated and the arduino is not isolated and therefore you can't get a common ground.
What you need to do is to look at your signal on an oscilloscope, that will tell you what is going on.
Another way round this is to have a voltage to frequency converter on the plasma end feeding it's output to an opto isolator going into an arduino pin where you measure the frequency and hence the voltage.