A solicited proposal is when the customer asks for a proposal. They may ask verbally or they may issue a written Request for Proposals (RFP). An unsolicited proposal is when you send them a proposal they haven’t even asked for because you think they should buy from you or take some action.
Solicited proposals are usually sent to customers who issue an RFP. When a customer wants something that is too complicated to pick up at the store or order from a vendor, they often write down a description of it and issue it as an RFP. If it is a commodity, they may issue a Request for Quotations (RFQ). An RFQ usually requires minimal information and a price. An RFP may require an extensive description of your approach or offering, as well as its price.
A solicited proposal provides you with a description of what the customer wants. Many also provide you with formatting instructions for your proposal and the evaluation criteria that will be used to make a selection.
Sometimes you will make a suggestion to a potential customer and they will ask you to submit a proposal so that they can consider your suggestion. This counts as a solicited proposal because they are expecting it and you have a chance to talk to the customer and gain an understanding of their needs.
An unsolicited proposal is sent to a customer who has not requested it. Unsolicited proposals must be especially convincing since the customer has not anticipated, planned, or budgeted for the proposal. With an unsolicited proposal you run the risk that the customer won't even bother to read it, since they didn't ask for it. However, the lack of competitive pressure with an unsolicited proposal often makes up for the risk.
If you send the same unsolicited proposal to a bunch of customers, what you really have is a brochure and not a proposal. A proposal should take into consideration the customer's specific environment, needs, and concerns. Proposals have a much higher win rate than brochures, because they better align what you offer with what matters to the customer.