Oooh, where do I start. Disclaimer to begin with: I run
http://cycle.travel/map which is
clearly the best route-planning website there is (ahem), so I may be more than a little biased...
Bike route-planners are essentially rules-based. For each road, they calculate a weighting, which is distance multiplied by a quality factor. So 10km of quiet lane will get a better score than 10km of motorway, and so on. The basis of this is usually road classification, but other factors such as surface quality can be taken into account. The route with the best weighting overall is the one that's then presented to you. So the trick is to find a route-planner whose weightings accord most closely to your style of cycling.
RideWithGPS uses Google route-planning (i.e. exactly the same as the bike mode in Google maps) if you have the Google background map switched on. If you have an OpenStreetMap-based background map, it switches to an open-source program called Graphhopper which uses OpenStreetMap data. The Google-based routing is often tolerable, sometimes terrible. The Graphhopper routing is slightly better but I wouldn't say that either produces particularly brilliant results. There are lots of other good things about RWGPS, but the auto route-planning isn't their strongest point.
Cyclestreets uses an entirely custom routing engine they've written themselves, again based on OSM data. You can choose from three sets of weightings (quiet, balanced, fast). The balanced routing is very good - one of the best I've seen. The downside is that the user interface is a little old-fashioned - you don't get a big map display, you can't drag the points around, and so on. But it'll almost always find you a good route.
Garmins have their own little built-in routing engine, but the processing power available in a Garmin is a fraction of that available to the online route-planners, so don't expect too much.
Which leaves cycle.travel and Strava. Both of these two are based on OSM data but augment it with additional information.
For Strava, this is popularity with other cyclists. For cycle.travel, it's motor traffic levels (where such data is available, which means the UK and the US). Put broadly, Strava is the best choice if you think a good route is defined by "lots of other cyclists", whereas cycle.travel works if your definition of a good route is "not many cars".
There is no right answer - it all depends what sort of cycling you enjoy. Personally I find planners like Strava direct me onto busy roads too often (e.g. asking for a route from Oxford to Chipping Norton sends me along the B4044 over the Eynsham toll bridge - no chuffing way am I cycling that), which is why I built cycle.travel to prioritise quiet roads. But for some people it's more important that you get your head down and the miles in, so Strava will be fine.