To quote the practice manager of one of the hospital's I've worked at: "We're fed up of taking on teenagers who stand in the corner, faint at the first sight of blood, don't help out at all and say they're bored". If you want to stand a chance you have to stand out by demonstrating that you aren't just another 15/16 year old who suddenly decided come work experience week that they quite like the idea of becoming a vet and playing with animals.
Make sure you detail any relevant work experience in your letter to demonstrate that you are committed/organised/mature/professional/etc. If you haven't already done a couple of weeks on a dairy farm, a couple of weeks at a kennels/cattery and a couple of weeks at a stables/riding school get that first and foremost. You should find it much easier to secure these sorts of placements, you are going to need to get this experience eventually and most practices will be more receptive once you have that experience.