Long story short, there is no black and white answer to the question of if you should run by distance or time. Correctly implemented as part of a training plan, both will result in performance improvements. Personally, I like structure and being able to measure my progress as I run, so prefer running by distance – but not all the time. Here are a few thoughts on each.
Race Distance
Lets say you are training for your first marathon and are using a generic plan. Running by distance you have a 20 mile long run, running by time you set aside 2.5 hours (Coach Jack Daniels cautions against beginners setting long runs of over 2 – 2.5 hours duration (the time that an elite athlete would complete a 20 mile training run) as the stress of running the 5+ hours 20 miles would take would be too much).
Here’s the catch … In 2.5 hours an elite marathoner running would have completed the distance, running 10 min/mile I would be at mile 15, and a new runner at a 15 min/mile would have completed 10 miles. Would you feel 
prepared to run a marathon with your longest run being 10 miles? I wouldn’t be, but as a coach there are ways that I can help you increase your mileage and preparedness based on a 2-3 hour long run.
Pacing
Most races are measured by distance, e.g. 5k, half marathon, full marathon. At some point during training you need to figure out if you are ready to complete your race distance, and also the pace that you can expect to run. The best way to do this is to simulate your race by running by distance during training.
Running by distance allow you to look at your pace, seeing what it feels like and how it changes over the race distance. If you have a time goal looking at your pace during training will show how you run (e.g. do you go out fast and slow down through the distance, maintain a steady pace, or go out slow and accelerate) and provide the opportunity to work on race strategies.
Workout purpose
Running by distance you may find yourself focusing on time rather than effort. You keep checking your watch, pushing your pace to finish earlier. The problem with this is that each workout in your training plan has a purpose, designed to help you reach your race goal. Workouts include “easy” runs to help build endurance and aid recovery after more intense “effort sessions”. If you keep pushing your pace, you will change the intention of your easy runs, negatively affecting your overall training and increasing your risk of injury.
Running by time gives a greater incentive to stick to the prescribed effort for that session, as time is time, you won’t achieve the session goal by running faster! But … if you tend to go easy on yourself you risk running too slowly, what should be an effort session (e.g. tempo run) into an easy run, missing out on mileage and effort.
Talking of effort sessions – I break my training into phases, following the training pyramid developed by the great Arthur Lydiard. These phases involve race specific workouts including hill work, tempo runs, and intervals, each with a specific purpose geared towards race day. You could complete all these workouts by time or by distance, however when you look at the intent of the workout it makes sense to use a combination. For example, a tempo run, aimed at improving your lactate threshold level, should be at a “comfortably hard pace” (threshold pace) of around 20-40 minutes. Running by time you get a workout that is appropriate for your fitness level and that you can maintain. If you have a distance goal your time at threshold pace may be too short, or not maintainable.
Injury
This is the big one for me. Coming back from injury it is important to go by feel, letting muscles, tendons and bones 
gradually adapt to the stresses of running, until you are at a point that you have a good cardiovascular base and can run easily for about 30 minutes. During this period it is important not to be focused on performance, avoiding comparisons with pre-injury pace and risking re-injury by pushing too hard.
As an aside, if you are sick missing a couple of days training is not going to affect your race performance. Once recovered, simply pick up where you left off without trying to make up lost miles for the week. Obviously, if injury has kept you away for a longer period you should re-evaluate your training.
Scheduling
Running by time you know precisely how long you’ll be out running, unlike running by distance where factors such as the weather or simply being tired and running a slower pace than usual are factors. The downside to running by time is that routes are often out and backs, which can be boring for some. If you have a busy schedule, easy to set time aside and not risk skipping training runs because you don’t know if you could get the mileage completed.
As I said at the outset, there is no right answer to whether you should run by distance or time. I use both as follows:
Easy runs: distance
Tempo runs and hillwork: time
Intervals: distance
Long runs: Road-distance, Trail-time
Injury recovery/prevention: time
Cross training: time