In short, management is about managing the day to day activity of the company (day to day activities include HR management, reporting, procurement, issue handling, conflict management, research, etc...). Project management on the other hand, is about managing project using a standardized process (often referred to as a project management methodology).
In small companies, the role of the functional manager might be exactly the same as that of the project manager, in addition to other tasks (such as logistics, employee evaluation, strategic management, etc...)
A functional manager can manage projects, but obviously, in your case, you're not managing projects according to a methodology, you're just getting some work done in a random, non-organized way (which technically cannot be called project management).
As for the last part of your question, the project manager does lack authority over the resources, especially in functional organizations. Trust me, when you will hire a project manager in your company, you will see that this is true, in fact, you will make sure that this is true (because otherwise you will feel that you will lose control over your resources to the project manager).