The Advanced Tools in the TravelTime toolbox provide the most configurable way of using the three core tools - Time Map, Time Filter, Routes.
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The core parameters of any of the Advanced Tools are the same as in the equivalent ArcGIS Simple Tools. Under the Advanced dropdown, however, are a number of additional fields that can be used to adjust the most granular aspects of the analysis.

​Public Transport Boarding Time - the time required to board a public transport vehicle
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​Walking Time - the max time for walking from the origin to the first public transport stop, and from the final public transport stop to the destination (value applies separately to both)
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​Driving Time to Station - the max time for driving from the origin to the train station (only applies to driving+train mode)
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​Cycling Time to Station - the max time for cycling from the origin to the train station (only applies to cycling+public_transport mode)
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​Parking Time - the time required to park a car or bike (only applies to driving+train and cycling+public_transport modes)
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​Ferry Boarding Time - the time required to board a ferry
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These Advanced Parameters are by default populated with reasonable values. These same values are also used in the background when running any of the Quick or Simple tools.
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Search Range is by default disabled. If a value is entered, however, Search Range will be enabled and a range of results will be returned (if applicable) based around a range of arrival/departure times.
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For example, if an arrival time of 9am is chosen, with a Search Range Width of 60, the tool will return results that have an arrival time between 8am and 9am. The same search with a departure time of 9am, will return results that have a departure time between 9am and 10am.
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This aggregation of multiple works slightly differently for each of the tools:
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Let's take a look at how adjusting the Advanced Parameters effects the results we get. The easiest way to do this is using the Time Map Advanced tool.
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Below shows the results of running the tool for a 1 hour reachable area around a number of car dealerships in Johannesburg, with Result aggregation set as UNION and the Advanced parameters set at their default values:

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If we increase the Walking Time value to 45 minutes, here is the difference:
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The reachable area has expanded for two reasons:
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1. Routes that involve only walking due to lack of access to any public transport (such as those around the dealership south of Randfontein) are now capped at 45 mins instead of 15 mins
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2. Journeys from public transport stops/stations that were more than 15 mins walk away from a dealership can now achieved, making whole new areas reachable

If we now set the Walking Time back to 15 minutes but instead increase the Search Range Width to 60 minutes, here is the difference:
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With a Search Range Width of 0, a Departing Time of 10am, and 1 hour Travel Time, the reachable area shows any journeys that start and finish between 10am and 11am.
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With a Search Range Width of 60 mins (and the same other parameters), the reachable area shows any journeys that start between 10am and 11am, and can be completed within 1 hour. So for example, a journey could begin at 10.55am and take 45 mins, arriving at the destination at 11.40am.