Separation pay is paid for non Chapter 61 discharges. From DoDI 1332.38:
If a member not entitled to physical disability compensation under Chapter 61 of reference (b) due to the circumstances listed above has six years or more, but less than 20 years of active service, immediately prior to that discharge, the member shall be entitled to separation pay as an involuntary discharge or release from active duty as prescribed by Section 1174 of reference (b), unless the Secretary concerned determines that the conditions under which the member is discharged do not warrant payment of such pay.
Severance pay is for separations under chapter 61 for ratings of 0,10 or 20%.
Severance pay is usally twice as much as separation pay.
Mike