I have both, but I spent less than half your budget to get them. That being said they are both on the small side.
My Yanmar 1401D does not have a loader on it but it does a great job tilling the garden and pulling a small trailer. It is scary on hills and I have rolled it before. It however is much lighter (1000 lbs) than many other tractors available. It however is low hours, completely reliable, and hard to get stuck. It is very rough on the yard and gravel driveway though as the rice and cane tires on it will dig deep to find traction. I paid around $2000 for it with several attachments.
I recently bought a New Holland L425 skid steer for $3000. It also is a very small skid steer, which gives me very good visibility in all directions compared to a Bobcat S150 I rented before. For such a small machine it is incredible what it can lift. It actually is not hard on the yard or driveway as long as I don't turn tight. My tires are completely bald and foam filled. The foam fill makes the machine much more stable and I don't hesitate to run all over hillsides with it. I run chains on all tires for traction and unless I get in soft wet clay or very deep loose dirt, it always has enough traction to go. I would also much rather roll a skid steer than a tractor. However if you remember where the weight is and keep the boom low, a skid steer is a stable machine. Weight kits and liquid or foam filled tires are must have items on a skid steer. My particular skid steer though is down now with a engine/starter issue. It was a pain to get the boom out of the way to access the engine (only a problem on the design of mine, most skid steers you can access the engine regardless of boom position) every thing is easy enough to work on. It is much easier than having to split a tractor for sure.
If you need a PTO, have general farm work to do, or want to pull a trailer, log splitter, or even skid logs, the tractor is the better choice. The cost and parts available for the tractor are better also. Even finding implements is much easier and less expensive since the 3 point hitch is universal. If you want to do loader work you need four wheel drive and the bigger the tractor, the more useful it is. However unless you run turf tires a tractor will always tear up turf.
If you want to do loader work or move dirt the skid steer is the hands down winner. It is much more maneuverable and can lift much more than a tractor of equal cost. If you make tight turns on a skid steer you will tear up the yard in a hurry. If you learn to make large radius turns though a skid steer can be gentle on the grass. The cost of parts and implements for a skid steer is higher. Also it wasn't until the mid 90's that the quick connection became universal so new buckets require modifications to fit old equipment. As others have said it is a pain to get in and out of a skid steer and they are very rough rides. It also takes lots of patience to get good at operating a skid steer. You can get many of the same attachments for a skid steer as you can for a tractor but they cost more. I would however recommend going small with a skid steer. A small skid steer is more maneuverable and has better visibility than a large one. For the most part a small skid steer can do the same job as a large one. It just takes longer as the smaller unit will have to make more trips to haul the same amount. But the big unit with its 84 inch bucket won't fit between the trees or go thru the barn door.
