If you were handed an Equity contract tomorrow, you would need to read the associated rule book cover-to-cover. You don’t need to memorize it, but you need to have a sense of what it contains so you can references it later. Then if you have questions about a particular rule, you can post them here on SMNetwork!
The specific rules for each contract dictate the ways in which the production process for an Equity show will be different from a non-eq. The rules about safety and scheduling are among the most impactful to SMs in terms of the daily flow of rehearsals & show calls. I don’t think anyone can give you a comprehensive list of other differences because things vary so much between theater companies. A lot of the differences I’ve encountered stem from budgets, space, and staffing. While Equity houses tend to have greater resources, it isn’t always the case, especially when comparing a low-paying contract, an SPT-1, for example, with an established non-equity company.
Weekly submissions may vary by contract also. In SPT for example, the SM does not provide anything to Equity on a weekly basis. The SM packet provides a number of documents that you use only if a specific situation occurs, and they are submitted as needed. I’ve never looked that closely at what the deputy submits, because one of the great things in my mind about having a deputy is the fact that I don’t need to cover their responsibilities.