Both have their place in the workshop, but there really isn't anything like watching that needle swing across the face of a dial, on the case of wiggling needles, i always admired the old style VU meters on the front of older style hi fi units, leds just don't cut it.
Amen to that, brother.
Just to keep things in perspective, there certainly are situations where an older analog meter just won't do. Anytime a high input impedance is needed, in order not to overload the circuit being measured, the tens of megohms input of a DMM is clearly superior to the tens (or hundreds) of Kohms for the best of analog meters.
Other than that, old-school mechanical meters, with their D'Arsonval movements, give a much clearer indication of magnitude than a bunch of LED or LCD digits.
By the way, I neglected to mention the "hybrid" that actually satisfies both requirements (easy-to-read analog display and high input impedance): the old-fashioned VTVM (or a newer solid-state meter with FET front end). This is the best of both worlds, and I highly recommend getting one.