They are different categories of things - survival analysis is the analysis of data where time to a given event is the dependent variable. The given event may include death, failure of a machine, a criminal's time to (re)offending or becoming ill, for example. It uses a number of techniques to analyse this data, including certain generalised linear models, including sometimes for the purpose of analysing whether specific variables influence the probability of an event, logistic regression.
Overall, survival analysis encompasses many techniques and methods to achieve different subordinate objectives, including tools for exploratory data analysis, distribution fitting and methods for designing experiments. Hence, I don't think you can meaningfully say 'build a model using survival analysis' rather than, perhaps 'build a model using Weibull regression/ Cox regression', as examples of tools closely associated with survival analysis.
Logistic regression, on the other, is a regression technique for analysing binary data. Hence, it is a single tool (but still a very powerful tool useful in many contexts) rather than an overall category of analysis. It could possibly be described as one of the central tools of categorical data analysis, where categorical data analysis belongs to the same category as survival analysis.