I need to create 5 methods on the server side, which will work with binary data. The remote clients are applet and JavaScript. The Client will send files to the server, and
Well, I wouldn't agree with your colleagues' opinion that isn't good to have rest used by only one application, since you may decide in the future to have different applications using the same rest api. If I were you I would choose the pure REST. Why?
If you're using some framework for rest implementation (let's say apache cxf or jersey) you get lots of stuff out of the box - you write POJO you get a rest, you get serialization and deserialization, to and from let's say, JSON object out of the box (eventually you will need to implement some JsonProviders but this is not a big deal).
It is intuitive to work (if you design your rest APIs well).
Very easily consumable by JavaScript clients (especially if you're using JQuery or something like that)
However, it strongly depends of what exactly do you want to do, if you have some strong transactional logic, the rest could be quite tricky. If you're only going to do POST requests (without using the other HTTP methods) you might want to use the Servlet since you won't have to work with additional frameworks and making more dependencies. Note that the REST is more or less an architectural concept and it does not contradict with the Servlet technology, if you're stubborn enough you can make a rest api only with servlets :-). Hope I've helped.