Well, in a nutshell - yes it makes a difference but you can keep it simple and still get a great result. The raw ingredients you choose will directly affect the qualities of your finished bar of soap. And this is why making your own soap is so good. You can tweak it to suit your skin type. Or the intended purpose of your soap.
Lets take coconut oil as an example. Coconut oil soap gives big bubbles and lathers incredibly well. It is is said that fishermen used to use it as it would even lather in sea water ( and salt is a big lather killer). It is also incredibly deep cleaning which can also make it drying - so most of the time you would include some coconut oil for it's positive qualities but balance it with other oils. It also makes a very hard bar of soap.
Olive oil gives less lather but is extremely mild to the skin. Soap with a high % of olive oil will also be very soft and require a much longer cure time.
Cocoa butter is conditioning to the skin and also contributes to the hardness of the finished bar.
In my soap making workshops we use a recipe combining all 3 of the above and this creates a very simple but well balanced bar of soap.
But the list of potential oils and butters you can use are endless and all soap makers will have their favourites. Caster oil is extremely popular and helps contribute to a fluffy and stable lather. It has an unusual fatty acid profile and is quite unique. It would generally only be used in small percentages (less that 5%) however as it can leave your finished product with a tacky texture in higher %.
Mango butter is one of my favourites, it does not contribute towards lather so I would keep the percentages relatively low but it is moisturising and luxurious. I once used it in a 100% coconut oil soap with a 20% superfat ( more on that here ) And then 2% of mango butter on top of that using the hot process method so I retained the mango butter as a superfat, it made a beautiful bar of soap.
Palm oil is similar to coconut oil but less drying, it contributes to bubbles and hardness of the bar. It does have bad press due to the environmental impact and I chose from day one not to use it but it can be sourced responsibly so do not immediately discount it.
And so the list goes on, there are numerous resources out there that will give you a huge long list of butters and oils you can use but I do not intend to list them here. I am all about 'soap making made easy' and making it accessible to newcomers. You do not need a complicated recipe full of numerous ingredients to make a lovely bar of soap that you can use on a daily basis. It just needs to be balanced.
If you would like to learn more about how to make soap with my online soap making video course, eBook or book one of my soap making workshops in Dorset you can find out more here