Time delay: digital low pass filter VS analog low pass filter
There are many kinds of filters, and the transient (time-domain) response is directly related to the frequency-domain response.

But it doesn't matter whether the implementation is analog or digital; any filter with a given frequency response is going to have the same transient response.

You pick a filter design based on which aspects of its performance are most important in your application. Some architectures have especially flat frequency-domain passbands, others have particularly steep transition bands, and some are optimized specifically for their time-domain response (e.g., no "ringing"). It's a topic that's really too broad to address here.