When you start a business, there are generally expenses incurred prior to being "open for business". Things like purchasing equipment, buying advertising and the such. Those expenses incurred prior to being open for business are start-up expenses. For start up expenses it does not matter in what year they were incurred either. Some businesses can take several years to acquire all the business needs before that business can "open".
Start up expenses can be claimed in the first year the business is open and earning income. Even if the expenses incurred exceed the first year's income you still claim them all. Start up expenses get amortized and deducted over the first 15 years of business. If I recall correctly (and I'm not sure on this) you'll see those expenses listed as an amortized asset in the Business Assets section of the program.
An amortized asset is usually material and non-material costs incurred prior to opening the business. Expamples of this would be some forms of advertising your grand opening, or the cost of training your employees prior to opening. These costs are flat out deductible business expenses that can be deducted from business income. But they are amortized over the first 15 years of business with a portion deducted each year, because it's not uncommon for a business to not have sufficient taxable income in that first year to deduct all of those start-up costs in one fell swoop. But the tax rules do give you the option to deduct a maximum of $5000 of startup costs in the first year, with any remaining startup costs amortized and deducted over the next 15 years.
A capitalize cost is not a full on deduction "per-se". It's a requirement to depreciate the cost of hard assets (like physical equipment used in the business to produce income) Most business assets, such as a building to house the business, or to house inventory are depreciated over 39 years. That means you can deduct the cost of that hard asset over 39 years of business. However, that deduction is not permanent. Whenever you close, sell or otherwise dispose of the business in the future, or if you sell or otherwise dispose of a business asset, all of that depreciation taken on those assets has to be recaptured in the year of disposition and taxes paid in the year of sale, closure or other disposition on all that prior depreciation taken in years past.
So in conclusion an amortized asset is one whose cost is permanently deducted from the income that asset produces. But a capitalized asset is one that is depreciated over time and the depreciated value is recaptured and taxed in the year that capitalized asset is no longer used in the business.