There's quite a bit of difference, actually.

Size of the Negative

Medium format is a somewhat encompassing term. When one talks about 35mm (135 film), it's most often a camera that shoots a 36mmx24mm frame. There have been specialty cameras that use the 135 format to shoot other sizes, but most 135 cameras shoot this fairly standard sized frame.

Medium format, on the other hand, could be of the 6x4.5, 6x6, or 6x7 type. They all shoot the 120 film but produce images sized at 56x41.5mm, 56x56mm, and 56x67mm, respectively.

I'd say that these are the most common, though the film has been used in cameras that shoot up to 6x24!

Why size matters: The 35mm long side is 36mm. An 8x10's long side is 254mm. One has to double the size of the negative 3 separate times (36 -> 72 -> 144 -> 288) to get to this. Unlike the process of blowing up a digital image, where interpolation adds more pixels, blowing up a negative means simply spacing out the grans into a larger area. At some point, the image simply falls apart. (The medium format, on the other hand, only has to jump a little more than twice [56 -> 112 -> 224].)

You may not think that one doubling of size will make a difference, but keep in mind that each doubling degrades quality. The larger the format, the less doubling has to be done in order to make a large print.

Size of the Camera and Resolution

Because 135 needs to be blown up so much, camera and lens designers have been forced to seek lenses with incredible resolving power. They really do milk as much detail out of the world and onto the film as they possibly can. Still, there are limits to how much detail can be resolved in the small space of the 135 frame.

The medium format camera, having to hold a bigger negative, is obviously larger. Rangefinder type 120 cameras benefit in size because of the lack of mirror, SLR types can be quite big compared to the 135 cameras.

But, because the negative doesn't need to be blown up as much, the lenses are not taxed quite as much to resolve absolutely everything. Don't get me wrong, medium format lenses are superb. But, optical designs can get a little simpler. (Take this to the extreme and look at the optical design for a lens designed for a large format 8x10 camera)

What is the difference, as an a amateur, between using a medium format and a 35mm camera (and the photos you get)?

As an amateur, gear acquisition can be quite costly. Some medium format items, because of the lack of market, are coming very far down in price. The Pentax 645 system especially is very cheap to obtain.

To me, because the larger negative is blown up less, some film and developer combos open up to you. For example, I don't think anyone in their right mind would use Delta3200 with Rodinal - but shot at a lower ISO (around 800) in 120 - the grain structure, to me, becomes very pleasing.

Developing it is easier. No can to crack and the spool doesn't require scissors to clip the film off. Simply unwind, remove the backing paper, and load.

You will get less frames per roll which has both pro's and cons. Some medium format cameras have the ability to swap film backs mid-roll - which is a definite plus over 135, IMO.

But, perhaps the biggest difference you would see is in the image quality of your blown up prints. There is a difference at 8x10 levels but medium format, hands down, beats 135 above that. If you aren't printing any bigger than 4x6 or 5x7 - you won't notice a difference.