This question is very wide, And I can only give my 2 - cents regarding one specific problem.
As long as you are in the US - there is usually no problem ( speed, or otherwise ) but in other countries - that is a different story.
Regarding CDN - In some countries, those CDN´s will not work well ( if at all )
One such example is China - where I often travel to on Business. And in China, as you know they have the "Great Chinese firewall".
Well - due to recent very published quarrels between Google and the Chinese government, Google services are often blocked, partially or totally. Google has even closed its CN domain and moved it to HK .
What does that mean? It means that websites that rely on google web fonts, or CDN ( for jQuery for example ) might not work in China, or will have sporadic unpredictable effects.
As a first result - their loading time will be 10s of times slower ( if at all ) and usually will end with a time-out on some parts .
The same effect is happening ( and not many people pay attention to that ) with any embedded twitter, or facebook codes. The pages just won't load.
Now, you can say you do not care about China, and that this is not your target audience - but IMHO disregarding a third of the world's population is a bit strange.
More over - This has happened to me in OTHER COUNTRIES AS WELL ( granted - most are totalitarian regimes, but still ...)
Regarding google - fonts - you did not really elaborate on the project, but one of the reasons I do not use them frequently is that most of them support ONLY English characters (Note: not LATIN, ENGLISH). So unless you specifically search for them in the font - a lot of European characters will not render ( like ß, á ,À ,í ,ü,ä,é
just to name a few but the actual list is much longer ) - not to mention other languages ( Arab, Hebrew , all Cyrillic based languages like Russian, Bulgarian , Ukrainian - All Asian languages like Chinese, Japanese , Korean and basically any language that is not ENGLISH . Even Spanish is on the list with ¿
and ñ
... so here goes all the Latin in the US... )
So - if your "project" is something that is UI based, and it is meant to be used with Other languages in the future - this requirement is not weird at all...
Summing up - you should use google CDN and font only if you provide a valid fall-back
.
Read : Best way to use Google's hosted jQuery, but fall back to my hosted library on Google fail
Like said at the beginning - this is just my 2-cents on the matter. I am sure other people can give you a different point of view.