Hi Gregory,
Its not a backslash (), rather a forward slash (/) you are referring to

The forward slash indicates that the url is ended. This tells the search engine: the request stops here.

Search Engines as correctly said by Chris does not differentiate between url with forward slash or without it - both are same. However, its advisable to have url ended with with forward slash to speed up the loading. To validate this - refer this excellent article - http://webdesign.about.com/od/beginningtutorials/f/why-urls-end-in-slash.htm

See, if you going to change domain name and aptly make correct 301 redirect to corresponding pages (following the same url pattern) of new site with same **content, title and description - **then you are home.

Adding to what Mr. Robert said - you can also check on url hierarchy through Xenu Link Sleuth or Screaming frog seo spider of your old domain and new site (post migration). Compare the link structure. If everything goes on well - you may experience 301 redirect working upto 99% and can almost see your similar rankings with new domain