The product groups will work on deals over all industries but only on their product. So an analyst in the M&A group may work on a Tech acquisition, and then a merger in the Consumer & Retail space.
The industry and country coverage groups will work on all products ( M&A, ECM, DCM) but only within their industry. So an analyst in the FIG group might work on an insurance companies IPO, and then work on deal where a bank wants to acquire another bank.
The M&A product group tends be execution heavy, so they're the guys who will crank out most of the modelling & generic analysis but they wont have in-depth knowledge of a certain industry. On the flipside, the industry guys should know their industry like the the back of their hand. They also do some modelling, and they also go out and pitch for mandates (origination), whereas the M&A product guys dont go out and pitch.
ECM and DCM are also split into subgroups (Structuring, Syndication, Origination, Private Placements, Convertibles, Leveraged Finance etc). The origination/execution mix is different for different groups. For example, ECM Origination is very pitching-intensive, and you wont do as much modelling/technical work relative to other groups (which is why ECM Origination is perceived to not be so great for exit opps). On the flipside, Lev Fin is a DCM group and is pretty technical, so places well into Private Equity.
What happens on most deals is that the industry and product groups will work in conjunction with each other. For example, the Energy & Power group may go out and pitch to a company looking to make acquisition. Assuming theyget the mandate, they'll do all the industry-specific analysis and valuation, and analyse the best way to pay for the deal (through cash, debt or equity). This is where the product groups step in, and do most of the modelling & analysis regarding the payment structure.