Atw23-
You asked what the difference was between a synth and a keyboard so I think some clarification is needed of terminology.
A synth, strictly speaking, would be a device, with or without keyboard, that creates its own sounds through a number of possible different methods. These methods can include sample playback, wavetable, additive, subtractive, FM, analog, granular and a number of others, including combinations of the above. The definition has become very blurred because sample playback (the most common method right now, probably) isn't really synthesis, per se, as the sounds already exist in memory and, most of the time, what is really happening is that those samples are being manipulated in various ways through filters, amplifiers, envelopes and effects.
A keyboard could be lots of things. Most basically, it could be anything that HAS a keyboard attached to it which could include, organs, electric pianos, acoustic pianos, synthesizers, workstations, midi controllers (which don't produce any sounds at all) and probably a few other devices I'm not thinking of.
I believe you are probably thinking of a workstation like the Motif, Triton or Fantom which are the big three from Yamaha, Korg and Roland, respectively when you ask about a "keyboard". All of these units offer a huge assortment of high quality sounds which can be modified and manipulated in a multitude of ways, they include sequencers for recording and saving your creations and are expandable. All are capable of creating a complete composition, from start to finish, hence the "workstation" nomer. They all have various models covering a variety of price points for different budgets.
Which one is best for you? That's impossible for anyone but you to determine. They are all good, capable units though.
When people talk about "synths" now, I think, most of the time, they would be referring to analog or analog modeling devices such as the offerings from Novation, Clavia, Nord and some Korg boxes like the new
Legacy collection or
MS2000B, etc. Of course, there are many others, I'm just mentioning a few of the more common units you are likely to see.
If you are looking for bread and butter string and piano and organ, etc. sounds, then a workstation would be the place to start. A lot depends on your budget, as it always does, so start with what you're willing to spend, consider used equipment and go out and play as many different units as you can!
Good luck. I hope this admittedly simplified explanation helps a little.
Ted