PMBOK, pg. 22: "Processes that are performed periodically as needed. The process Acquire Resources is performed as resources are needed. The process Conduct Procurements is performed prior to needing the procured item."
ACQUIRE RESOURCES (PMBOK, pg. 328), "Acquire Resources is the process of obtaining team members, facilities, equipment, materials, supplies, and other resources necessary to complete project work." We tend to think of this process as bringing on developers. But this process is actually more vast because in the previous process Estimate Activity Resources, we listed out all the necessary project resources to complete the work. One thing in all my projects is to acquire and schedule (book) a conference room for project meetings. In estimating activity resources, I would note I need (1) meeting room. In Acquire Resources, I would actually book/secure meeting room 14A1C (14th floor, etc).
The procurement process is the formal process for obtaining resources that are not available to the project manager internally. The simplest way to think about this is if you are a PM for a construction company. The company doesn't keep raw materials on hand, but rather has vendors supply sheet metal, nails, etc. In terms of manpower, we could procure an architect or a software developer; this maybe true even if we had developers available in the company, but yet none of them had the requisite experience.
Acquire resources is about formally obtaining project resources, conduct procurements is more focused on contractually entering into agreements with vendors on how to satisfy needed project resources.
Hope this helps.
