Check if they have any trademarks or copyrights to protect their name, logo, image, etc.
No real problem in using the same name if both businesses are local and miles apart, especially if your websites etc look completely different. Lots of businesses do that.
Problems arise if you're both regional or national, where customers can easily get confused.
The major challenge is the legal term "passing off" which you should google to get a good idea of what it means etc. Basically it's where you're trading on the back of another business, i.e. pretending to be them, benefiting from their marketing/goodwill, etc.
At the end of the day, lots of businesses operate happily using the same names. Often because they're in different markets, i.e. selling different things, operating in different areas etc. You just have to be very careful not to breach any trademarks or copyrights they have and take all steps to make your business as different as possible from them, i.e. different logos, different website colour scheme, keep space between their locality and yours etc.