Why have an appraisal? 

Much of the sales data and listing data is difficult to obtain for an area unless access to MLS data is available.  
In a non public disclosure state like Texas, MLS data becomes even more important when sales prices of For Sale By Owner and Builder sales do not appear in court house records.  Popular internet valuation models are not able to use sales data from the court house since it is not available so they must rely on listing prices which can vary greatly from the final sales price.  Even with the MLS data, one might not be able to determine proper adjustments to make to the sales data of the comparables to give an estimate of value.  An experienced and qualified appraiser has both training and field experience in these areas.

Can I just apply the avg cost/sq. ft. for the neighborhood to the size of my house?  

If each sale is virtually identical to your home in terms of features such as size, quality, beds, baths, updating, condition, extra features (such as a pool), exterior appearance, garage type and size, and it is a recent sale, then price per sq. ft. might be somewhat accurate.   If not, then the price per sq. ft. multiplied by the subject property size can give a misleading value.   Generally, it is difficult to find many comparables that are recent sales and that are identical to the subject property.  An appraiser is trained to determine the appropriate adjustments for these differences to arrive at a reasonable estimate of value for the subject property.  Lenders and investors generally do not accept a value just based on price per sq. ft. and require generally accepted appraisal adjustments on predetermined appraisal forms for acceptance for a loan.