In full honesty, if you didn't get a scholarship for Melbourne, and you don't want to go to Monash because you didn't get a scholarship from them either, that's a pretty shitty reason. There are always more chances to apply for scholarships (and every year at the start of the year, Monash opens up applications again to apply for scholarships that didn't get handed out, so you could always apply after you get in and get a scholarship anyway), so don't let that be a deterrent.
Of course, if you DO have a scholarship for Melbourne, it's a completely different story.
As for the question asked, people in biomed scholars gets a scholarship just for being in the degree (so hey, scholarship anyway). Other than that, I don't think there's much difference between this and the regular biomed degree at Monash - but it may be similar to the old Science Scholars degree in that you didn't need to have done 24 units to get the degree (as long as you fulfilled all the requirements of it, still), as well as the nice common room available to those students (I've heard there's one for biomed advanced, not sure if it's also for biomed scholars). Having said that, are you sure it still exists? Because I can't seem to find it on the Monash website...