Verb[て] (manner/means)
Noun[で] (means/instrument)
The て-form of a verb and the particle で after a noun can both express means, manner, or method – how an action is performed or what is used to accomplish it.
When a verb in the て-form precedes another verb, it can describe how the main action is carried out. The て-form verb shows the manner, means, or accompanying action.
Here, the て-form verb describes the way the main action happens. This is different from the sequential use of て (where actions happen one after another); in the means/manner use, both actions occur simultaneously or the first verb explains how the second is done.
The particle で after a noun indicates the tool, means of transportation, material, or method used to perform an action.
Both structures express means, but they are used with different parts of speech. Compare:
In the first sentence, the verb あるく歩く (to walk) is in the て-form to show how you go. In the second, the noun たくしいタクシー with で shows what means you use. Both answer the question "how do you get there?"