I have two past writings that are useful here:
If you read both of those you have a good starting point about what I’m going to say. Value is what is found in accepting the argument being conveyed, but stakes are more personal.
For example, in a sale, if your product will save a company 20% on their costs, and those costs are $1 million, you can assign an accurate value of $200,000. If you’re selling to the owner, you can convey this value, and the owner can understand, but it is still different from the stakes. The stake is the owner’s livelihood. The business staying profitable and successful. His friends, family, and community seeing him as a continued success rather than one of the lucky ones for only a moment in time. Those are the stakes.
Value derives from the need.
Stakes derives from want, even the unconscious kind.
Make sure whoever you are persuading understands both.