There are no differences simply the second notation is not recommended by IUPAC. The explanation is below:

What is ionic mass? The ionic mass is the real mass of the ion that takes in account the weight of the electrons in the overall weight. However electrons are much more lighter (0.548 mDa) then the nuclei so in fact usually in spectroscopy is used the mass number, A: > Total number of heavy particles (protons and neutrons jointly called > nucleons) in the atomic nucleus. Also called nucleon number. Symbol m > in mass spectrometry. And not the ionic mass (that in fact would more accurate) hence the m symbol in the equation refer to the mass number.
 What do they mean by "z a multiple or elementary charge"? This is another approximation they suppose that we could use an elementary charge +1, +2, +3, -1, -2, -3 etc. etc. that is not a real quantification in Coulumb of the electrical charge of the ion as stated Jannis Andreska.
$\ce{C7H7^2+}$, m/z equals 45.5 -> what did they do here? They simply divide m (see definition above) by z in this case (2+=2).

Addendum: This definition silently says that the m/z ratio is an approximation m is not the real weight of the ion (in fact is the weight only of the heavier particles and z is not the real charge of the ion) so it says: "don't use 'e' because this is an approximation we are not using the real charge nor the real weight, and the m/z ratio is is a dimensionless quantity!". I believe that in the future when even the lighter electrons, due the increasing accuracy of the instrument, will matter we will use $m_{i}/q_e$ where $m_i$ is the ionic mass and $q_e$ is the real charge... Only matter of time (and matter!)!

Reference here!

There are no differences simply the second notation is not recommended by IUPAC. The explanation is below:

What is ionic mass? The ionic mass is the real mass of the ion that takes in account the weight of the electrons in the overall weight. However electrons are much more lighter (0.548 mDa) then nuclei, so according to IUPAC definition m is intend to be A:

Total number of heavy particles (protons and neutrons jointly called > nucleons) in the atomic nucleus. Also called nucleon number. Symbol m in mass spectrometry.

And not the ionic mass (that in fact would more accurate). 

What do they mean by "z a multiple or elementary charge"? This is another approximation they suppose that we could use an elementary charge +1, +2, +3, -1, -2, -3 etc. etc. that is not a real quantification in Coulumb of the electrical charge of the ion as stated Jannis Andreska.
$\ce{C7H7^2+}$, m/z equals 45.5 -> what did they do here? They simply divide m (see definition above) by z in this case (2+=2).

Addendum: This definition silently says that the m/z ratio is an approximation m is not the real weight of the ion (in fact is the weight only of the heavier particles and z is not the real charge of the ion) so it says: "don't use 'e' because this is an approximation we are not using the real charge nor the real weight, and the m/z ratio is is a dimensionless quantity!". I believe that in the future when even the lighter electrons, due the increasing accuracy of the instrument, will matter we will use $m_{i}/q_e$ where $m_i$ is the ionic mass and $q_e$ is the real charge... Only matter of time (and matter!)!

Reference here!

What is ionic mass? The ionic mass is the real mass of the ion that takes in account the weight of the electrons in the overall weight however electrons are much more lighter (0.548 mDa) then the nuclei so in fact usually in spectroscopy is used the mass number, A:

Total number of heavy particles (protons and neutrons jointly called nucleons) in the atomic nucleus. Also called nucleon number. Symbol m in mass spectrometry.

And not the ionic mass that in fact would more accurate, hence the m symbol in the equation.

 What do they mean by "z a multiple or elementary charge"? This is another approximation the suppose that we could use an elementary charge +1, +2, +3, -1, -2, -3 etc. etc. that is not a real quantification in Coulumb of the electrical charge of the ion as stated Jannis Andreska.

 $\ce{C7H7^2+}$, m/z equals 45.5 -> what did they do here? They simply divide m (see definition above) by z in this case (2+=2).

Maddendum: This definition silently says that the m/z ratio is an approximation m is not the real weight of the ion (in fact is the weight only of the heavier particles and z is not the real charge of the ion) so it says: "don't use 'e' because this is an approximation we are not using the real charge nor the real weight". I believe that in the future when even the lighter electrons will matter we will use $m_{i}/q_e$ where $m_i$ is the ionic mass and $q_e$ is the real charge... Only matter of time (and matter!)!

Reference here!

There are no differences simply the second notation is not recommended by IUPAC. The explanation is below:

What is ionic mass? The ionic mass is the real mass of the ion that takes in account the weight of the electrons in the overall weight. However electrons are much more lighter (0.548 mDa) then the nuclei so in fact usually in spectroscopy is used the mass number, A: > Total number of heavy particles (protons and neutrons jointly called > nucleons) in the atomic nucleus. Also called nucleon number. Symbol m > in mass spectrometry. And not the ionic mass (that in fact would more accurate) hence the m symbol in the equation refer to the mass number. 
What do they mean by "z a multiple or elementary charge"? This is another approximation they suppose that we could use an elementary charge +1, +2, +3, -1, -2, -3 etc. etc. that is not a real quantification in Coulumb of the electrical charge of the ion as stated Jannis Andreska. 
$\ce{C7H7^2+}$, m/z equals 45.5 -> what did they do here? They simply divide m (see definition above) by z in this case (2+=2).

Addendum: This definition silently says that the m/z ratio is an approximation m is not the real weight of the ion (in fact is the weight only of the heavier particles and z is not the real charge of the ion) so it says: "don't use 'e' because this is an approximation we are not using the real charge nor the real weight, and the m/z ratio is is a dimensionless quantity!". I believe that in the future when even the lighter electrons, due the increasing accuracy of the instrument, will matter we will use $m_{i}/q_e$ where $m_i$ is the ionic mass and $q_e$ is the real charge... Only matter of time (and matter!)!

Reference here!