I have recently found old (very old) laptop and though that it might be great opporunity to learn assembly and some basic OS development.

Assembly tutorials are most often dedicated to the 8086 chips, so there my question is, does this Intel 82340SX make any difference? Does anything change regarding language usage?
The 80386 chip is mostly (but not completely) backward compatible with 8086. For the differences, see the relevant section of Intel’s 80386 Programmer’s Guide (e.g. http://www.scs.stanford.edu/nyu/04fa/lab/i386/s14_07.htm ).
The 80386 also has new features that 8086 didn’t support. A few important features are protected mode, paging and support for 32-bit code. I strongly recommend researching these features; as without them your software will be limited to slightly more than 640 KiB of RAM (the extra 3 MiB of RAM reported as “extended memory” will be wasted) and there will be no effective isolation between pieces of software (e.g. applications can trash each other and read each other’s data). If you do this right, software designed for “80386 only” will be able to use about 3 GiB of RAM on new machines without any major changes.