Well, the functions do different things.
First, there are two internal implementations of date/time: POSIXct, which stores seconds since UNIX epoch (+some other data), and POSIXlt, which stores a list of day, month, year, hour, minute, second, etc.
strptime is a function to directly convert character vectors (of a variety of formats) to POSIXlt format.
as.POSIXlt converts a variety of data types to POSIXlt. It tries to be intelligent and do the sensible thing – in the case of character, it acts as a wrapper to strptime.
as.POSIXct converts a variety of data types to POSIXct. It also tries to be intelligent and do the sensible thing – in the case of character, it runs strptime first, then does the conversion from POSIXlt to POSIXct.
It makes sense that strptime is faster, because strptime only handles character input whilst the others try to determine which method to use from input type. It should also be a bit safer in that being handed unexpected data would just give an error, instead of trying to do the intelligent thing that might not be what you want.