The words ἁνάπαυσις and, ἄνεσις, are both translated ‘rest,' but there is a difference in their application.
ἀνάπσις. (from ἀναπυω, ‘to give rest') is appropriate to refreshment after labor or fatigue, and also to an intermission of action, and is variously applied. It is the word often adopted in the LXX for the ‘rest' of the sabbath. The Lord invited all that labored and were heavy laden to come to Him, and He would give them 'rest' (ἀναπαύω). Also to take His yoke, and such should find ‘rest.' Matt. 11:28, 29. The unclean spirit walks through dry places, seeking ‘rest,' but finds none. Matt. 12:43; Luke 11:24. In Rev. 4:8 the four living creatures ‘cease' not day or night in their ascriptions of praise; and in chap. 14:11 Those who do homage to the beast will have no ‘respite' to their sufferings.
ἄνεσις (from άνίημι, ‘to loose') is more the relaxing of cords or bonds, or rest from tribulation. Felix ordered the centurion to let Paul have ‘liberty' and that his friends might visit him. Acts 24:23. Paul when waiting for news of the Corinthians, as to the effect of his first epistle to them, had no rest' in his spirit until Titus arrived. 2 Cor. 2:13; 7:5. With regard to the collection for the poor saints, Paul said he did not mean the Corinthian saints to be burdened, and others to be ‘eased.' Chap. 8:13. Paul also tenderly speaks of the saints at Thessalonica being delivered from their tribulation, to ‘rest' or ‘repose' with him and others at the revelation of the Lord Jesus. 2 Thess. 1:7.