One of the other important considerations not mentioned so far, is solder melting point, especially for those of us working on older (pre- 2006) equipment, the bulk of which will usually have been designed for, and produced with, a leaded solder.
The 'normal' 60/40 or 63/37 leaded solder products, used extensively before lead-free solder legislation, are the lowest melting solders - while, as far as I'm aware, all of the lead-free and silver containing solders are higher melting point products. The higher temperatures involved with lead-free products may seem 'trivial' and easily handled by modern, often temperature controlled, soldering irons / stations, but the potential problem lies in the higher thermal stresses applied to older pcb traces, that were not designed for the higher melting solders. Excessive temperature can easily lead to trace damage, delamination, etc. - and while pcb quality obviously varied over the years, and between different manufacturers, few if any of the earlier pcb's were designed for the elevated temperatures required by the lead-free products, so 'caution' is called for.....
Everyone will have their own preference, but personally, I tend to use desoldering braid to remove old solder, or initially a solder sucker for larger quantities, and then desoldering braid - I think it's not only effective at wicking away the old solder, but also provides a measure of thermal heat-sinking that helps minimise (potential) damage to pcb traces. I normally resolder using lead-based solders, for personal use, but understand that there may be different constraints for professional restorers.