There's a lot of difference in tone between a Strat and a Les Paul, due to a combination of factors. You'll hear a lot of differing opinions about which of these is most important, so this is in no particular order... and take it all with a grain of salt
First, the standard Strat setup is three single coil pickups, while a Les Paul has two hum buckers. This factor alone has a huge impact on tone. Single coils are generally brighter, while humbuckers have a big more 'oomph'. You can find Strats with hum buckers installed, and you can get Les Pauls with 'coil taps' that emulate the sound of a single coil, but neither is a true emulation of the other.
Second the bridge construction is much different. The Strat generally uses a floating tremolo and bridge, while the bridge on the Les Paul is mounted directly to the body of the guitar. Some say this has a tone impact. In practical terms, there's generally no 'whammy bar' on Les Pauls, though you can find them with Bigsbys, etc.
Scale length. The Strat has a longer neck, which means the strings are a bit more taut, which contributes to the tone. Each will play a bit different because of this as well.
Body Design. The Strat is usually made of ash or another similar wood, often in three parts. There are a ton of different Strat varieties out there however. The Les Paul is often made of mahogany with a maple cap, which many say has an impact on the tone. I personally notice body wood difference more with acoustics.
The guitar necks are different too. There's the scale length mentioned earlier, but generally the classic Strat design includes a quarter sawn one piece maple neck, which is attached via bolts to the body. This was made for easier swapping for working musicians. Due to this variety, you can often mix and match necks on a Strat easily. The radius on a Strat neck can often be smaller than a Gibson, making them feel different when bending and requiring different setups. The Les Paul has a neck design that is prone to breaking at the headstock, and is made of mahogany (I might be wrong here) and rosewood traditionally. It's set directly into the body and glued in permanently - that's what 'set neck' means. Depending on the fretboard radius, it may be easier to bend and be able to handle lower setups. I think that fret board wood will have more of an impact than body wood on electric guitars.
I personally wouldn't get a bolt on Les Paul unless I truly just wanted the look at a particular price point. I'd either save for an Epi - there's plenty of options from them in a variety of looks - or throw some PAF-esque humbuckers in my Strat. In my opinion, while all of these factors contribute to each guitar's unique sound, changing pickups will have the most immediate impact on your tone.
