My local Apple shop suggested for $39 I upgrade to Snow Leopard. What differences will this make? I don't keep up so have no idea.

Tiger is 10.4.xx

Leopard is 10.5.xx

Snow Leopard is 10.6.xx

Looks like you've got Leopard if it's 10.5.8

If everything's working fine for you now, then there's no need to upgrade. It's just something else to worry about, and a way for Apple to suck ya dry. Don't sweat it, you'll be O.K.

Snow Leopard is more stable, performs better and is better supported than Leopard. Unless you must stay with 10.5.8 for whatever reason, I would upgrade. Plus 10.6.6+ gives you the Mac App Store. Make sure you have at least 2GB of RAM before moving to Snow Leopard.

Tiger is 10.4.xx

Leopard is 10.5.xx

Snow Leopard is 10.6.xx

Looks like you've got Leopard if it's 10.5.8

If everything's working fine for you now, then there's no need to upgrade. It's just something else to worry about, and a way for Apple to suck ya dry. Don't sweat it, you'll be O.K.
Looks like I'm out of RAM! I'll boost it to 4GB as recommended in another post about me complaining of slow performance

Snow Leopard 10.6.x still receives regular security patches (at least until Mountain Lion 10.8.x comes out later this year).

Leopard 10.5.x no longer receives significant security updates.

That being said, Apple's product support lifecycle for its software is not as clearly defined as a certain other popular computer operating system vendor. Some high profile security vulnerabilities may cause older software to receive some attention after Apple has otherwise stopped paying attention to an OS.

The recent Flashback Trojan security patch is actually prime example of this situation: Snow Leopard and Lion both received updates which fundamentally repaired the underlying vulnerability which allowed Flashback to be a problem in the first place.

Leopard, on the other hand, only received a token attempt to remove any existing infestations, and a rather trivial patch to make it harder to accidentally pick up the infection.

Other publicly revealed vulnerabilities have also received fixes in Snow Leopard
and newer, but never received any attention at all on Leopard.

Bottom line: If capable of running it, your Mac would be more secure after upgrading to Snow Leopard than it currently is running Leopard.

In a couple of months after Mountain Lion is released, the same argument will apply to persuade you to ditch Snow Leopard in favour of Lion -- but the bar to upgrade to Lion was set a little higher due to its reliance on 64-bit CPUs.

Looks like I'm out of RAM! I'll boost it to 4GB as recommended in another post about me complaining of slow performance

You're currently out of RAM in that shot because you'er running too many applications and processes at the same time.

And yes, 1GB of RAM is pretty minimal.