I’m a headhunter in the financial services industry, so perhaps I can be of some help here. “Sales and Trading” refers to people who are in one of two different business lines: Fixed Income or Equities. Fixed Income includes bonds, foreign exchange (currency) and commodities, and all kinds of funky derivatives thereof. Equities essentially means stocks, and all kinds of derivatives thereof. So those are, in broad strokes, what you would be either selling or trading, depending which of the 2 sides of the business you went into.
There are modelling people in both Fixed Income and Equities, and they often come up with trading models, i.e. strategies for how to trade certain products given a slew of variables, such as what might happen with interest rates, with yield curves, with currency prices, etc. There are also modelling professionals who focus not on trading models, but on structuring complex deals/transactions that are done for customers who want to accomplish something complicated, such as a corporation or an insurance company that wants to make its balance sheet look more attractive, with less “risk” on it, having hedging strategies in place in case certain commodities or investments go south, etc. So modellers are very mathematical, quantitative types of people who crunch a lot of numbers and work a lot with computers, as opposed to interfacing with clients like, say, a salesperson would do.