I have used ApexSQL for a few years now and I have also used various RedGate tools. Both are superb offerings, but I have a pretty big soft spot for ApexSQL... simply because they offer free SQL Tools. When I was starting out as a dev, ApexSQL offered me great tools without the price tag. I found them via this link way back when http://dirkstrauss.com/apexsql-tools-free/
When I started working as a contractor, being able to use some free tools kept my costs down. So ApexSQL definitely has a space to compete.
SQL Code Compare and SQL Search are my top go-to apps. In fact, the SQL Search has been indispensable when working on large databases that I was unfamiliar with.
If I had to choose between Redgate and ApesSQL, I wouldn't. I would use both as needed.