Hello,
I'll try to explain this as short and to the point as possible, hoping I don't fall too quickly into the trap of technobabble.
Anyway, first I want to state the thing both formats have in common: Both MP3 as SHN are audio compression formats. A regular audio track (shown on a computer as a WAV file) is about 10 Megabyte per minute. Both formats can compress this audio into smaller files.
The difference is that MP3 compresses the audio by leaving out certain data/sounds. This is done mostly in the upper and lower range of the sound spectrum (the very high treble and very low bass). By doing this it can have a compression rate of about 1/10th of the original track. As a consequence, it isn't possible to get back to the original through decompressing the MP3 file, it cannot restore the data it left out. This is why MP3 is called a 'lossy' format; it loses information.
In contrast, SHN compresses the audio track without leaving out any data. It may replace certain oft-repeating patterns with a shorter substitute string and use some other tricks, but it will not delete anything. As a consequence the compression rate for SHN isn't as high as the compression rate for MP3; it'll reduce the original audio track to roughly 2/3 of its original size. But the advantage is that by decompressing the SHN file you get an exact replica of the original track. This is why it's called 'lossless'; there is no loss of data.
Is this clear?
Let me give a short example...
Say you have a page full of text and you want to compress it. By replacing oft-repeated strings (example: the words 'the' and 'you') with a shorter string (for simplicity, here with '1' and '2', although in reality those strings will be unique). This will reduce your page of text. In addition, the MP3 method will leave out rarely-used characters (like an 'x' and an '#'), shortening the page even further. However, you never get these characters back, leaving you with an incomplete (but still readable) page once you decompress.
OK, that's my attempt to explaining it.
Hope it helps...
C ya!
Marty