While none of these alternative sites are perfect replacements for Classic Maps, they're what we've got for now. I haven't checked most of these out personally yet, but I'm linking to them anyway in case you find them helpful for your purposes.
General Maps Based on Classic Maps
Note: Google has recently changed their API so that all maps now have the new-style controls. However, this doesn't affect most of the underlying features.
This site is designed to let people make additions and corrections to Google Maps. However, it uses Classic Maps as its base map and provides some of its features including Directions (under the Edit menu), a variation of Street View/Maps split screen, and 2D satellite view.
An excellent alternative to Classic Maps with some of its most popular features, this mapping solution has undergone some changes since first developed. Check out this post for the most recent update (01/18/16).
Google Maps Streetview Player of Brian Folts
A reader emailed me this link with some detailed instructions about how to use it. There's not enough room for me to write them up here, so I've copied them into a comment on the original Google Maps post.
Instant Street View
Thanks to reader Sappy for sharing this find that has a lot of Classic Maps' features (searchable, zoom, 2D satellite view). However, it doesn't have a full split-screen Street View or directions.
Thanks to reader Szymon for sending along this link for a Polish site. The map has Street View (not split-screen) but doesn't have search or Directions.
Thanks to reader Dave for sharing this German site.
T4L reader Wareko is also developing a map based on Google Classic, available in Japanese and English!
Specialty Maps Based on Classic Maps
Thanks to reader James for sharing this website that he maintains to help people find tennis courts around the world.
Thanks to reader S.H. for sharing this Google-based map that includes solar radiation and temperature data.
Thanks to reader John for sharing a whole slew of URLs that incorporate Classic Maps.
Maps Based on Other Platforms
One of the original mapping applications.
Microsoft's mapping site.
Yahoo Maps Directions
While Yahoo! has shut down its main Maps site, this implementation is currently still working.
Open-source mapping software. A reader adds, "OSM is just the database: It's up to someone to read the data and build actual maps, and possibly provide a router such as OSRM for directions."
Thanks to reader Metal for sharing this site, which describes itself as "a multilingual open-content collaborative map, where anyone can create place tags and share their knowledge." Has both map and satellite views, but does not include a street view or directions feature.
Thanks to reader Frederic for sending me these links. Descriptions in quotations are taken directly from the site.
Ride with GPS
"Ride with GPS makes it easy to drag and draw your own custom routes with lightning fast elevation profiles to help you anticipate (or avoid) those big climbs."
Bike Route Toaster
"BikeRouteToaster.com is a course creation application primarily aimed at Garmin Edge/Forerunner owners although other users without a GPS may also find it useful for planning rides."
Pretty straightforward-looking route planner built with OpenStreetMap.
Ride with GPS
From their About page: "Ride with GPS was founded by two guys that love writing cool software and riding their bikes: Zack Ham and Cullen King."