Grants.gov v NIH ASSIST for grant submission
Functionally, you can use whichever system you prefer; there is no inherent advantage in submitting via either system, since both are approved methods for submitting proposals.

The decision on which system to use basically boils down to which system you feel is easier, and which one your university's grants and contracts administrators are more comfortable using. (The easier it is for them to use the system, the easier it will be for you to submit your proposals according to the rules and deadlines!)