There are few differences and some similarities.
In a typical software product firm, the product manager is accountable to deliver a product which leverages the firms distinctive competency and serves a large identified market.
You start with identifying a market problem, doing technology and competitive benchmarking, creating MRDs, PRD, working with engineering teams to build the product and then work with your sales and marketing team to launch and sell the product. Depending on the size of the market, breadth of the solution, your own experience in product management, and the role of product management in the company, the product manager drives the ‘internal’ build-out, road mapping and is responsible for the ‘product’ and the sales guy carries the ‘number’. Few orgs have the PM responsible for P&L (in my experience very few are structured this way though). You might also have other PMs or Analysts that you might manage.
In an eCommerce company, you typically have product management which is aligned to the various critical functions of the eCommerce life cycle.
PM’s are aligned to specific feature sets like Search, Navigation, Content, Cart and Checkout, Shipping, Social, SEO, API, Mobile etc. Each PM is responsible for their area. For example, the Search PM is accountable for internal search on the site, optimizing both the feature function and resulting revenue. In my experience (having been a PM in classical software product firm and eCommerce PM) a classical software PM typically is optimizing product feature function for market fit, while the eCommerce PM optimizes for increased conversion and customer satisfaction. In most eCommerce product management organization, I have seen product management to be more ‘business’ focused metric driven versus software product managers have stronger bias for ‘product’. In most parts, activities like problem identification, requirements definition, road mapping are similar. But you will not see much of persona build outs, win-loss analysis, channel training, sales training etc.
Also in some organizations, non-technical PM roles tend to be slotted into vertical businesses such as shoes, automotive etc (where it has more merchandising flavor)