1. You might use them both in numerous contexts and no one would think you were saying anything that differs much whichever you chose.

(TLFi) c) [Avec valeur intensive] De la manière la plus profonde, la plus authentique, au delà des apparences.

• Les élèves sont vraiment intéressés par les travaux pratiques.

2. There is however at least one subtle différence when “vraiment” is better taken au pied de la lettre; the intention is not any more to give added intensity to what the verb communicates but to discard a doubtful impression or answer to a direct question as to whether or not a real interest is involved; the translation of “vraiment” is then “really” or “genuinely“. Of course, again, in this case, “beaucoup” can be used to the same effect of giving a positive answer that will have a similar consequence, but the use of “vraiment” will be decoded as meaning “really”, not specifically “much”.

Case 1

• — It seemed you might want to go half and half on that deal with me, I haven’t talked to you about it for some time.

• — I’m really interested, do not mistake my silence about it for a lack of interest.

• — Il me semblait que vous seriez d’accord pour faire moitié moitié sur cette affaire, je ne vous en ai pas parlé depuis quelques temps.

• — Je suis vraiment intéressé, ne prenez pas mon silence à ce sujet pour un manque d’intérêt.

Case 2

• — Can we suppose that this scheme of two monthly consultations could appear ideal to them?

• — I think they are really/genuinely interested in settling for it as soon as possible.

• — Est-ce que l’on peut croire qu’il trouvent idéal cet arrangement en deux séances de délibérations mensuelles ?

• — Je crois que ça les intéresse vraiment de l’accepter aussitôt que possible.

3. When the negative form of this locution is used, a similar interpretation is the best: “not really”.

(TLFi) Nég. + vraiment. [Signifie que l’on se refuse ou que l’on hésite à affirmer que]

• They pretend to like nature, but it doesn’t really interest them.
• Ils prétendent aimer la nature mais elle ne les intéresse pas vraiment.