tl;dr: You'd probably see the most difference with how a) the data is distributed and replicated and b) how you query data.
CouchDB (as of the 1.5 release) offers master-master (including offline) replication. It does not offer sharding. Cloudant's BigCouch does implement a dynamo-like distribution system for data that is slated for CouchDB 2.x iirc. Mongo on the other hand does support sharding via mongos, and you can build replica sets within each shard. It does not as far as I know support master-master. This is probably the biggest data-distribution difference between the two.
MongoDB support a more SQL-like ad hoc querying system, so you could query for drink recipes with 3 or less ingredients that have vodka in them, for instance. You'd still need indexes on the data you are querying for performance.
CouchDB queries are facilitated via javascript or erlang map reduce views, which serve as indexes you craft. An additional 'secondary-index' like query facility is to use a lucene plugin and define searchable data. Cloudant has this baked into their offering, and their employees maintain the plugin on github (https://github.com/rnewson/couchdb-lucene)
MongoDB has the ability to do things like append a value to a document array. In Couch, you'd likely read the entire document, append to the array in your app, and put the document back on Couch. It does have an update functionality that can sometimes isolate things more than this, but I haven't seen it used as much. Mongo can also do things like increment counters, while Couch cannot (though CouchBase can).
There's a host of other differences. Mongo has a much broader API, while CouchDB takes a more simple http verb like approach (get, put and delete see the heaviest use). Depending on your situation, one might be a better fit, or you might simply grok one more than the other.
As far as why Mongo gets used more often, I think the closer-to-SQL ad-hoc queries made more sense to people transitioning from stores like MySQL. The CouchDB view/map-reduce stuff is a bit more of a mindset shift (see the View Collation wiki entry for an example of this at http://wiki.apache.org/couchdb/View_collation). CouchDB was also taken hits for being slower than Mongo, but I suspect it was the map-reduce stuff that really steered some folks the other way.
Marketing. They shipped with unacknowledged writes for a long time and it made them look really good in write benchmarks. Couch was actually trying to keep your data safe. But it didn't look fast enough so those that didn't read the fine print on page 195 from the manual where it tells you how to enable safe data storage for MongoDB, jumped on the bandwagon.
Oh and mugs, always the mugs. I have 3 I think.
> I'd like to get a picture about why Mongo seems to be used more often than Couch.
Very good marketing from 10gen on the one hand. On the other, CouchDB is older (and we techies love the new hotness), and the CouchDB/Couchbase split confused a lot of people. Having your original founder found a different and incompatible project with almost the same name but very different goals would cause almost any project to stutter.
Yes, that really didn't inspire confidence in the longevity of the project
If you're doing primarily insertion with querying via id, and using views in which stale data is ok, then couch is far superior to mongo. But that's not a use case everyone has.