To allow is to permit or acknowledge something, the way a teacher might allow a student to be absent, or an insurance company might allow a claim. While not necessarily an enthusiastic show of support, allow implies no effort to hinder something from occurring. Acquiesce, on the other hand, is full of reluctance. To acquiesce is to assent tacitly, or to submit or comply silently or without protest. Typically when a person acquiesces to or in something, they're not thrilled about it, but after objection, debate, or negotiation, they're accepting it with a degree of resignation.