LSJ wrote:κατάκειμαι, Ep. 3 pl. κατακείαται Il.24.527: Ion. plpf. κατεκέατο Hdt.7.229; subj. -κέωμαι Pl.Smp.213b, Cypr. part. ka-ta-ki-me-na κατακιμένα Kafizin270 (sense 3, 225/18 B.C.):—Pass., only in pres. and impf. with fut. Med. -κείσομαι:—lie down, μῆλα τὰ δὴ κατάκειτʼ ἐσφαγμένα Od.10.532; ἐπὶ πλευρὰς κ. Il.24.10; νέκυς κ. Tyrt.11.19; ἐφʼ ἁρμαμαξῶν μαλθακῶς κατακείμενοι Ar.Ach.70.
b. lie in a grave, IG10(2).1.352 (Thessalonica, iv A.D.), SEG29.310 (Corinth, A.D. 529), ἐνθάδε κατάκιτε .. TAM4(1).353.
2. lie hid, ἐν λόχμῃ .. κατέκειτο μέγας σῦς Od.19.439; θάμνῳ ὑπʼ ἀμφικόμῳ κατακείμενος Il.17.677.
3. lie stored up, δοιοὶ γάρ τε πίθοι κατακείαται ἐν Διὸς οὔδει Il.24.527; τό γʼ ἐν οἴκῳ κατακείμενον ἀνέρα κήδει Hes.Op.364: metaph., ἄλγεα .. ἐν θυμῷ κ. Il.24.523.
b. to be deposited, of deeds in a registry, POxy.1040.32 (iii A.D.), etc.
4. lie sick, keep one’s bed, Hdt.7.229, Ev.Marc.1.30, etc.; lie in bed, Ar.Ec.313; ἐφʼ ὃ κατέκειτο Ev.Luc.5.25.
5. lodge, reside, Hp.Epid.1.26.ε ́, 3.1.γ ́, al.
6. to be idle, X.An.3.1.14; of things, lie neglected, καθεύδειν ἐᾶν ἐν τῇ γῇ κατακείμενα τὰ τείχη Pl.Lg.778d.
7. recline at meals, πῖνε, κατάκεισο Ar.Ach.985, cf. Hdt.3.121, Pl.Smp.185d, Ev.Marc.14.3.
8. of land, lie sloping to the sea, πρῶνες ἔξοχοι -κεινται Pi.N.4.52.
9. ἀρετᾷ κατάκειται πᾶσαν ὀργάν is expended in every impulse on .., Id.I.1.41.
10. to be given in pledge, Inscr.Cret.4.47.1, 10, al. (Gortyn, v B.C.); cf. κατατίθημι I 4 b.
Liddell, H. G., Scott, R., Jones, H. S., & McKenzie, R. (1996). A Greek-English lexicon (p. 893). Oxford: Clarendon Press.
Compare:
LSJ wrote:ἀνάκειμαι, poet. ἄγκειμαι, serving as Pass. to ἀνατίθημι, to be laid up as a votive offering in the temple, to be dedicated, κρητῆρές οἱ .. ἓξ χρύσεοι ἀνακέαται Hdt.1.14; ἀ. ἐν ἱρῷ Id.2.135; πρὸς τοῖς ἱεροῖς Lys.10.28: metaph., αἶνος Ὀλυμπιονίκαις ἄγκειται Pi.O.11(10).8, cf. 13.36; λόγος τῷ θεῷ ἀ. Pl.Smp.197e; ἐν οὐρανῷ παράδειγμα ἀ. R.592b; χρύσεοί κʼ ἄεκείμεθα Theoc.10.33.
b. to be set up as a statue in public, Σόλων ἀνάκειται παράδειγμα D.19.251, cf. IG14.1389 i 8, cf. Lycurg.51.
2. to be ascribed or offered, αἱ πράξεις ἀ. τινι Plu.Lyc.1; ἡ ἡγεμονία ἀ. τινι Id.Arist.15; ἐς τοὺς ἀστέρας τοὺς ἑπτὰ .. τὰς ἡμέρας ἀνακεῖσθαι D.C.37.18, cf. Polem.Cyn.15.
II. πᾶν or πάντα ἁνάκειται ἔς τινα everything is referred to a person, depends on his will, Hdt.1.97, 3.31: so c. dat. pers., πάντων ἀνακειμένων τοῖς Ἀθηναίοις ἐς τὰς ναῦς since they had their whole fortunes defending on their ships, Th.7.71; ἐπὶ σοὶ τάδε πάντʼ ἀνάκειται Ar.Av.638; ἅπαντα .. ἐπὶ τῇ τύχῃ μᾶλλον ἀ. ἢ τῇ προνοία Antipho 5.6; of persons, σοὶ ἀνακείμεσθα E.Ba.934.
b. w. εἰς, to be assigned or given over to, εἰς θάντον ἦν ἀνακείμενα τοῖς ἀογήσασι J.AJ17.6.5, λιμὸς εἰς ὑστάτην ἀνακείμενος ἀναισχυντίαν 18.1.1, εἰς τὸ σωφρονεῖν ἀνέκειτο ἡ ἐπιτήδευσις τοῦ βίου 18.3.4; so w. ἐπί, τὴν ἐπὶ τῷ θανεῖν ἀνακειμένην ἐπιστολήν 18.8.9, D.C.37.18; also w. dat., to be assigned to, Plu.Arist.15, Lyc.1.
2. to be put aside, ταῦτα ἀνακείσθω Them.in Ph.29.20.
III. lie at table, recline, S.Fr.756, Philippid.30, Arist.Cat.6b12, Fr.607, Diph.40 Mein. (om. Kock), Plb.13.6.8, Ev.Matt.9.10, al.; cf. Phryn.191.
b. to be laid up by sickness, PMich.XI 624.27 (vi A.D.).
Liddell, H. G., Scott, R., Jones, H. S., & McKenzie, R. (1996). A Greek-English lexicon (p. 107). Oxford: Clarendon Press.
So they are practical synonyms in part of the semantic range, but otherwise differ quite a bit.