128 bit 3DES Key and AES Key: what's the difference? Apologies if this is really basic but I couldn't find the answer to this.

I have heard some Info Sec colleagues talking about creating an AES key in preference to a DES or 3DES key but I don't know the difference; I thought a key was just a random string of bits? Is there a difference? Can you use the same key with different algorithms? i.e. a 128 bit key with 3DES and also with AES?
Both an AES-128 key (as defined by FIPS&nbsp;197), and a TDES Keying Option 2 key (as defined by FIPS&nbsp;SP-800-67) are 128-bit bitstrings. Similarly, both an AES-192 key and a TDES Keying Option 1 key are 192-bit bitstrings. The differences are:


In AES, all bits of a key matter to the result; in TDES, 1 bit out of 8 (the lower-order bit of each byte in the usual representation) either


is ignored, and can have any value;
is checked, and must be the complement of the exclusive-OR of the 7 others; that is, each byte is adjusted to odd parity; see this answer for details on the oddity (pun intended) of DES key bytes.

TDES has a complementation property, which  removes an extra bit of security for a given key size in many situations (including CPA attack); that further reduces a 128-bit TDES Keying Option 2 key to 111-bit strength, and a 192-bit TDES Keying Option 1 key to 167-bit strength.
The block size is 128-bit for AES, 64-bit for TDES; this means that AES and TDES are not interchangeable, even though they use the same key (analogy: truck and car are not interchangeable, even though both might run on gasoil). As rightly pointed in that other answer, that block size difference also makes a security difference in some applications.
AES is unbroken for any practical purpose, but there are attacks on TDES (at least, with Keying Option 2) using conceivable amounts of computing power and not-far-from-conceivable amounts of memory.
I'm leaving alone the non-issue of DES weak and semi-weak keys, which is immaterial in practice, and much more for TDES.


If one generates a TDES key with knowledge that 1 above will apply when it will be used, one can generate the key at random just as for an AES key (there might be different regulatory requirements). If one generates a TDES key with knowledge that 2 above will apply (or does not have such knowledge but want to err on the compatible side), one must adjust the low-order bit of each byte according to the 7 other bits.

One can use a TDES key for AES (AES-128 or AES-192 depending on TDES keying option); however, if the key was generated with low-order bit adjustment, that weakens AES-128 by $2^{16}$ w.r.t. brute force key search, which is (somewhat) undesirable and (very) unusual (AES-192 would be weakened by $2^{24}$, which likely is still strong enough except perhaps from a certification standpoint).

One can use an AES key for TDES (assuming matching key option) if


the TDES implementation does not check parity (case 1 above);
or the AES key is parity-adjusted to transform it into a TDES key.


In any case, it would be a bad idea to use the same key for both AES and TDES, for a successful key recovery attack on either cipher (e.g. by DPA) would then translate into an attack breaking the other.