The words "discretionary" and "quantitative" make it sound like one form of analysis uses math while the other is more whimsical. That's definitely not the case, both are mathematical and both are powerful tools.
Though both are useful, they are very distinct from each other so most firms are either composed of discretionary analysts or quantitative analysts but rarely both. If a firm has both the teams, they occupy distinct functions.
Discretionary analysis is mostly done using spreadsheets. For example, a discretionary analyst with use techniques such as discounted cashflows, debt structure, asset depreciation, market growth, etc. when evaluating the value of a possible deal. Often the model is built so that several scenarios can be considered. Like all modelers, discretionary analysts use assumptions to predict the value of assets. Robustness is usually done by varying the values of various assumptions to determine how sure the returns are.
Quantitative analysis uses statistical methods to approximate returns and variability. Quantitative analysis usually relies on datasets which makes it less effective (and often ineffective) when considering one shot opportunities. For example, quantitative finance isn't particularly useful when evaluating if a firm should participate in a merger or acquisition because these decisions are very dependent on circumstances. For example, you're not going to find a good training set that would be informative in deciding if Facebook should buy Instagram.
Since discretionary analysis doesn't rely on historical data the same way quantitative analysis does, that's what makes it "discretionary" and not "quantitative". Quantitative analysis tends to be more testable than discretionary analysis. There's lots of historical stock data to check if a trading strategy is profitable and the only thing required is to switch out datasets to test the strategy in different contexts but switching out Instagram from Snapchat in our discretionary analysis example would likely require a unique approach, not just switching out some cell values in Excel.
Typically, discretionary analysis and quantitative analysis are so different that there isn't a "discretionary way" and a "discretionary way" of approaching the problem. It'd be like saying when are nails the best vs when should I use glue. Sure they're kinda similar but they're so distinct that there isn't much overlap.
Perhaps an example where both approaches can be used is when determining what stocks to hold in your portfolio. Quantitative analysts will fit models on historical data and use them to predict returns and volatilizes in the future. Models and strategies such as CAPM, Black-Scholes, smart beta, and alpha research fit under the quantitative analysis umbrella.
On the other hand, a discretionary analyst might visit companies that don't have much analyst coverage and then use unique insights to determine which stocks to weight more and which to weight less in their portfolio. "Fundamental analysis" would also fit under the discretionary analysis approach.
Due to the constraints on quantitative analysis, if a unique situation is going to be analyzed, discretionary analysis is the only way to produce a useful result.
If a large amount of data is available and the situation isn't unique, then quantitative analysis may be a better way to go.