While both may have many overlapping features and can help with measuring performance and making projections, one simple answer is not applicable to every business operation.
Accounting software can provide data management for accounts payable, accounts receivable, general ledger, credit card processing, sales and work order processing, bar coding, workflow mapping, inventory management, returns, payroll and so forth. Basically, all the normal business accounting functions.
While a cloud or in-house accounting software package may not have the bells and whistles of a complete ERP solution, it may be more than enough if all you need is accurate reporting of financials. If you aren't handling major manufacturing, complex inventory systems or other advanced financial transactions, then an accounting software will probably be all you'll ever need.
Enterprise Resource Planning Software
An ERP can support an enormous range of financial tasks. You can use an ERP to record customer transactions, plan product rollouts, manage suppliers, track inventories, and keep all your usual financial records.
Most businesses that need an ERP solution usually have to integrate their human resource records with their financial data, customer and shipping transactions, inventory, warehouse, and supply chain information--all under a single umbrella. One central database allows most ERP's to track information for customer relationship, time management, quality control, and financial reporting/projections.
So Which Do I Buy, Lou?
Ultimately, it depends on what you need. If all you need is to your track your money well-enough so you avoid getting completely skinned at tax time, then a good accounting software will do the trick. However, if you are looking for deep information analysis about your customers and vendors, then an ERP is the better choice.
Whatever you do, don't just review your current needs! You must take into account the projected growth of your business and try to imagine what your needs will be in 5 years time. Any financial software you buy will be with you for a long time. Make sure you choose one that will give you the most flexibility down the line -- not just the one that's cheapest today.