These two terms refer to the same objects in two closely connected fictional universes. The first applies to Marvel Comics and the second to the Marvel Cinematic Universe. The Infinity Gems/Stones share the following characteristics in both universes.
- There are six of them.
- They are gems.
- They existed before the universe began.
- They comprise Power, Reality, Mind, Space, Soul, and Time.
- In fact, they're basically the same things.
The movie and comics versions have the following differences.
- They may have different origins, as discussed on this site.
- They have different names.
- They have, of course, been involved in different events.
Further, the MCU is designated Earth-199999 in the Marvel Multiverse numbering scheme, suggesting that the MCU constitutes a separate universe within the same continuity as the Marvel Comics multiverse, and thus that Infinity Gems and Stones may be exactly the same objects.
I have suggested infinity-gems as a synonym for infinity-stones, but there are not many active users with sufficient reputation to vote on it, with only 10 users having sufficient reputation. Therefore, it seemed best to seek community consensus.