An assignment is a transfer of ownership of something from X to Y. The thing can be physical property or intangible (eg the copyright in a script or software, the design of a chair or a patent on an invention – intellectual property rights). The parties involved can be individuals or businesses. Debts and loans can also be assigned. The power of an assignment is that you as the assignee become the owner.
With a license, there is no transfer of ownership; the licensee merely gets permission to certain usage which, without that permission, would be unlawful. Licenses can be as restrictive or as free as they are negotiated: they might be limited to a certain territory (just the UK or the world and universe); they could be for a set period (5 hours or 5 years). The licensee may have to pay an upfront fee and/or royalties. It may be just granted to one person for their exclusive use (exclusive license) or for more than one party (non-exclusive license).
One very important point to consider is that for an assignment to be effective it needs to be in writing and to state what is being assigned and signed by the owner (the assignor). This is the same for an exclusive license. So, even if you and the other party are on the understanding that you have an assignment it will only be at best a non-exclusive license if the above is not followed. Not good.
Therefore, if you want an assignment, then a handshake is not enough.