The main difference lies in when the abilities can be used.
Players have to be dead before Reincarnate and Rebirth can be taken advantage of. All the raid saving spells last week were designed to help prevent shiitake mushrooms from hitting the fan. But it's nice to know that if players do start dropping, your leaders have some options available. Everyone likes to have an ace up their sleeve.
There are good times and there are bad times to using combat revive spells. It's difficult to list every possible situation and even then, it's a matter of subjectivity among your raid leaders. I'll list a few things to consider when you decide to whip out Rebirth.
Use it only if you have to. The thing with these two cooldowns is that they have a long recharge timer. Personally, I don't like using them on the first few attempts of a boss that we're learning. I'd rather have a player stay down but remain observing the various aspects of the encounter. Using Rebirth on a phase 1 of some boss before we have any idea of what phases 2, 3 and 4 look like doesn't seem like a rational idea to me. Once the raid group has experienced all parts of the fight, that's when a Rebirth is considered and handed out. When going for the kill, your raid leaders might wish to stave off a Rebirth until later on in the fight (for example, during a phase 3 instead of a phase 1). Reasoning behind that is to wait and see if the raid can hold it together that long before bringing in the reinforcements. If the raid's going to buckle and collapse, then holding the Rebirth is a smart idea.
Consider raid needs. So your raid has some dead players. What's the res priority? Assess the situation first. If it's an endurance type fight and you're short healers, it might be wise to pick them up to reduce stress. If there's an enrage timer involved, snap up a DPS as their damage is going to be necessary to help speed up the process. On any tank sensitive fights, they become the priority. Kologarn is a good example of this. Let's assume one of the tanks get taken out. One tank can hold the line for some time, but unless the second tank is picked up, the raid will become overrun.
Always pickup Druids first (unless their Rebirth is on cooldown). The rationale behind this is that you can set up chain Rebirths among Druids in your raid. Druid 1 picks up Druid 2 who picks up Druid 3 before picking up that Mage, for example. It's a no-brainer, really. But just make sure the Druids that are being revived still have Rebirth's available otherwise you will wish to rethink who to cast on.
Timing matters. This is one of my biggest pet peeves. It has nothing to do with the Druids per se. When a player receives a Rebirth, they receive a pop-up box saying that they've been revived (Specifically, it's something like "Come back to life: Accept or Deny"). Hitting accept spawns the player at the point where the Druid was when they pulled off the spell. Now what really annoys me is when the player accepts the revive and then gets smacked down moments later by an AoE or some other spell. I understand the element of the random number generator. As in, random events can happen. But not everything is random. Boss abilities and cooldowns can be predicted. Going back to the previous Kologarn example, wait for his Left Arm to sweep and damage the raid before popping back up. Wait for Kologarn's eyebeam to lance out and target another player. Having the patience to wait and recognize when to activate the revive is something that some players need to work on.
Buff and heal them fast. Although not really a factor, it's still something which has to be done. When players get a Rebirth, they're going to be extremely weak. Heal them up as quick as possible before slapping them with whatever quick 30 minute buffs you can. I'd say the priority buffs are Power Word: Fortitude and Blessing of Kings.
There are a couple of addons that help keep track of certain cooldowns. Note that these only track cooldowns that are relevant to the entire raid. I doubt there is a huge amount of interest from people who want to know when my Penance is ready to go (About 7 seconds, for those that do want to know).
oRA2: The staple addon of raid leaders everywhere, oRA2 has a lot more functions than simple cooldown tracking. But for our purposes, that's all we're going to worry about. Last time I checked, Reincarnate didn't show up in the combat log. This means that Shamans have to have oRA2 set up and ready to go in order to transmit to the raid leader that they're back up.
RaidCooldowns: I just picked up this addon recently and haven't been able to conduct a full field test of it on my own just yet. It doesn't quite track a player's individual cooldowns (as an example, it won't track a Mage's mirror image). It goes after cooldowns that are relevant to the entire raid. For example, it'll track Rebirth, Innervate, and Divine Hymn among others. Certain abilities and spells can be hidden based on your preference.
Want some more advice for working with the healers in your guild? Raid Rx
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