I'd like to challenge a premise of your question:

I noticed that snow jackets tend to be cheaper than soft shell jackets and that they are waterproof and wind resistant, so I was wondering what issues I would face if I use a snow jacket?

You are putting too much emphasis on labels: "Snow jacket" is an arbitrary label that a company (seller or producer) may use, but it is essentially irrelevant to you. On the other hand "soft shell jacket" is a reference to a class of material used to make a jacket, so that's somewhat relevant. It's wrong to believe that's a uniform class of jackets, as in your case an "expensive" one. There's really cheap and crazy expensive shoft-shells.

What you should put emphasis on is characteristics: Is a jacket water-proof, how breathable is it (this is not a binary condition!), how much (if any) insulation does it have, ... How exactly this influences they choice of jacket depends both on the target conditions and personal preference.

For your conditions and my preference (items of clothing that can be used as versatile as possible, and are dependable, enduring and reasonably priced), I'd go for a non-insulated, water-proof jacket. That is usually labelled "hard-shell", but I wouldn't be surprised if you found something with only very light insulation termed a "snow jacket". This won't be very breathable, but who cares in rain, and outside of rain you can open it or take it off -> very breathable. And for >8degC any kind of insulation below the jacket that you probably already have (pullovers or whatever) is fine.

TLDR:
Ignore labels like "snow-jackets", look for characteristics.
That said many so-called "snow-jackets" will not be water-proof and have too much insulation for your use-case, but there might just be one that fits it perfectly.

I'd like to challenge a premise of your question:

I noticed that snow jackets tend to be cheaper than soft shell jackets and that they are waterproof and wind resistant, so I was wondering what issues I would face if I use a snow jacket?

You are putting too much emphasis on labels: "Snow jacket" is an arbitrary label that a company (seller or producer) may use, but it is essentially irrelevant to you. On the other hand "soft shell jacket" is a reference to a class of material used to make a jacket, so that's somewhat relevant. It's wrong to believe that's a uniform class of jackets, as in your case an "expensive" one. There's really cheap and crazy expensive soft-shells.

What you should put emphasis on is characteristics: Is a jacket water-proof, how breathable is it (this is not a binary condition!), how much (if any) insulation does it have, ... How exactly this influences they choice of jacket depends both on the target conditions and personal preference.

For your conditions and my preference (items of clothing that can be used as versatile as possible, and are dependable, enduring and reasonably priced), I'd go for a non-insulated, water-proof jacket. That is usually labelled "hard-shell", but I wouldn't be surprised if you found something with only very light insulation termed a "snow jacket". This won't be very breathable, but who cares in rain, and outside of rain you can open it or take it off -> very breathable. And for >8degC any kind of insulation below the jacket that you probably already have (pullovers or whatever) is fine.

TLDR:
Ignore labels like "snow-jackets", look for characteristics.
That said many so-called "snow-jackets" will not be water-proof and have too much insulation for your use-case, but there might just be one that fits it perfectly.