Hi,
@Arndt: Please correct me where I'm wrong and please give additional info where I fall short (or don't know myself).
The android Brouter app can read your Osmand favorites and use these as start/end points and use them as waypoints.
It calculates a route and adds this a a brouter0.gpx track to your tracks folder with subsequent numbers in the same session. Note though that it overwrites the tracks again in the next use of the program: Be aware of that!
Osmand can use these gps files as a route as well: load the gpx track as a map layer, select directions and OsmAnd asks you to use the track as route. Or alternatively from the map: select same start and end point and Osmand asks whether it should use the route/track.
Brouter uses navigation profiles (of course) which are actually simple text files and can be modified. They reside in /brouter/profiles2. This also means you can create your own profiles (if you know what to do :) ).
Brouter also uses its own "maps" where you can download full and car maps (cross), full maps (bottom-right to top-left) or car maps (bottom-right to top-left). This could use some explanation on the website and in the app. It took me some time to recognize it.I first thought it had something to do with slicing the map (only the top-left or bottom-right, or top-right or bottom-left, and which slice does it take?) of the map.
I tested the "fastbike" option and the "car-test" option and always compared them with the OsmAnd routing. I didn't test moped nor trekking.
I also added some images where the purple line is the OsmAnd route and the turquoise line is the Brouter gpx route/track.
"fast-bike":
(to start with: I don't know why it's called fast-bike as there is no "slow-bike" or "touristic-bike" or whatever, maybe as pendant to short-bike but that's not in the list.)
I tested it with routes up to 50-60 kilometers as that's about the maximum I do (I don't drive a racing bicycle. I want to enjoy nature, take pictures and a cup of coffee with apple pie every now and then: that limits time and distance).
I think it does a slightly better job then OsmAnd, but that's because Brouter uses cycleways more then "car" roads compared to OsmAnd. I prefer a nice cycleway above a road even if it is slightly longer (I don't know if its slightly longer/shorter as the distance differences mostly fall within a few 100 meters, br:osm: 20,8 vs 20,9 km (image zwolle-raalte-cycle-360x600.jpg), 23,3 vs 24,2 km, 43,3 vs 44,0 km, 83,9 vs. 79,2 km). It seems that the longer the distance the longer the route brouter uses, but it uses cycleways far better then OsmAnd which I prefer. It really calculates a cycle route instead of a route over roads where you can cycle as well (but that's my opinion).
So again: It's better in my world as I like to use cycleways over normal roads, e.g.: it's personal taste. I can't say anything about the "mathematical and algorithm wise" optimal route.
Occasionally you find some specific A*/Dijkstra examples (attached image zwolle-raalte-square-360x600.jpg) of algorithm culpits.
The "zwolle-raalte-cycle-360x600.jpg" shows the difference in routing which theoretically is only 100 meters in difference. The turqoise route is over cycleways, the OsmAnd first part of the route is over the normal road have a cycle-lane. Both take the same "touristical, scenic cycle route" for the second half.
I calculated quite some routes where Brouter always took the "nicer" route (again: my personal taste). I didn't make screenshots of all routes of course.
"car-test":
When doing a couple of calculations it turns out that brouter can calculate a long (600+ km) route where OsmAnd can't. It takes long to very long on longer distances, but it does calculate the route. It also calculates the same route as Osmand does with a few quirks occasionally.
When I calculated a route from Zwolle, NL to Sittard, NL, brouter routed me through 2 cities (image: zwolle-sittard-360x600.jpg) which is definitely shorter in km's but also definitely (much) longer in time. Same on a route from Zwolle to Riezlern (Au) where I had to use 2 waypoints for OsmAnd and did the same for Brouter. Brouter again routed me through a city being Kempten (image zwolle-riezlern-360x600), and then came back on the exact same route. Again: Osmand purple, Brouter turqoise.
(Brouter could calculate this route also without waypoints, but it took more then 10-12 minutes probably while it leads through Nordrhein-Westfalen which has an extremely dense road network).
But it is called car-test and obviously needs some (minor) corrections either in the algorithm or in the "car-test" navigation profile.