This is basically two context-dependent versions of Jaap's answer, each with a bonus:
Before running git diff --no-index, you may want to set diff.colorMoved in the Git config. Alternatively, you can include a --color-moved option instead of setting it globally.
diff.colorMoved settings use a different line color when lines removed from one place show up elsewhere in the diff... whether they moved within the file or moved to another file, like in a refactor.
Complete or almost complete sets
You have some files in a directory, current-version, that you want to compare to some other files. If the set of files you want to compare them to in new-version is nearly identical in content, just use Jaap's suggestion:
git diff --no-index current-version/ new-version/
Specific files to compare
If there are only a handful of files in one version and lots of files in the other, like so,
┌ current-project
| ├ (various folders)
| └ src
| ├ (various files)
| ├ foo.c
| └ bar.c
└ proposed-replacements
├ new-foo.c
└ new-bar.c
...you can try to let Git's rename detection do the work. I don't know what version of Git adds the degree of rename tracking that is necessary to do this (I'm using v2.18). Also, the testing I did was on a pretty trivial set of files.
You don't really want to see that the majority of the files are "missing" in proposed-replacements. Also, you don't want to bother giving proposed-replacements' files the same name/path as they appear in current-project, so you leave them where they were and put in a filter that includes only renamed and modified files.
git diff --no-index --diff-filter=RM current-project/ proposed-replacements/
diff --git a/current-project/src/foo.c b/proposed-replacements/new-foo.c
similarity index 65%
rename from current-project/src/foo.c
rename to proposed-replacements/new-foo.c
index dd51990..64445b1 100644
--- a/current-project/src/foo.c
+++ b/proposed-replacements/new-foo.c
@@ -1,4 +1,4 @@
octopus
cow
-dog
tiger
+flamingo
diff --git a/current-project/src/bar.c b/proposed-replacements/new-bar.c
similarity index 87%
rename from current-project/src/bar.c
rename to proposed-replacements/new-bar.c
...